Logic Programming with Prolog
Max Bramer
Logic Programming
with Prolog
Max Bramer, BSc, PhD, CEng, FBCS, FIEE, FRSA, ILTM
Department of Computer Science and Software Engineering
University of Portsmouth
United Kingdom
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Control Number: 2005923526
Apart from any fair dealing for the purposes of research or private study, or criticism or review,
as permitted under the Copyright, Designs and Patents Act 1988, this publication may be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing
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those terms should be sent to the publishers.
ISBN-10: 1-85233-938-1
ISBN-13: 978-1852-33938-8
Springer Science+Business Media
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© Max Bramer 2005
The use of registered names, trademarks, etc. in this publication does not imply, even in the
absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use.
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Printed in the United States of America
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SPIN 11344445
Introduction
Logic Programming is the name given to a distinctive style of programming, very
different from that of conventional programming languages such as C++ and Java.
Fans of Logic Programming would say that 'different' means clearer, simpler and
generally better!
Although there are other Logic Programming languages, by far the most widely
used is Prolog. The name stands for Programming in Logic. This book teaches the
techniques of Logic Programming through the Prolog language. Prolog is based on
research by computer scientists in Europe in the 1960s and 1970s, notably at the
Universities of Marseilles, London and Edinburgh. The first implementation was at
the University of Marseilles in the early 1970s. Further development at the
University of Edinburgh led to a de facto standard version, now known as
Edinburgh Prolog. Prolog has been widely used for developing complex
applications, especially in the field of Artificial Intelligence. Although it is a
general-purpose language, its main strengths are for symbolic rather than for
numerical computation.
The developers of the language were researchers working on automating
mathematical theorem proving. This field is often known as computational logic.
But if you are not a Computer Scientist, a logician or a mathematician do not let
this deter you! This book is aimed at the 99.9% of the population who are none of
these. Those who are, already have a number of excellent textbooks from which to
choose.
The idea that the methods developed by computational logicians could be used
as the basis for a powerful general purpose programming language was
revolutionary 30 years ago. Unfortunately most other programming languages have
not yet caught up.
The most striking feature of Prolog for the newcomer is how much simpler the
programs look than in other languages. Many language designers started out with
good intentions but could not resist, or perhaps could not avoid, making their
creations over elaborate. In some languages even writing the customary test
program to print out the words Hello World! to the user's screen is hard work. All
the user has to do in Prolog is to enter write('Hello World!').
vi
Introduction
Traditional programming languages have one feature in common. They all
contain a series of instructions that are to be performed ('executed') one after
another. This style of programming is called procedural. It corresponds closely to
the way computers are generally built. This is a tempting approach that has been
used since the 1950s but is ultimately flawed. The way users write programs
should depend as little as possible on how the underlying machine was built and as
much as possible on what the user is trying to do. In just the same way, the
facilities I use when I drive my car should depend as little as possible on how the
engine was designed or the carburettor works. I want all that level of detail hidden
from me, although in practice I understand that it may not be completely possible
to ignore it.
Prolog programs are often described as declarative, although they unavoidably
also have a procedural element. Programs are based on the techniques developed
by logicians to form valid conclusions from available evidence. There are only two
components to any program: facts and rules. The Prolog system reads in the
program and simply stores it. The user then types in a series of questions (strictly
known as queries), which the system answers using the facts and rules available to
it. This is a simple example, a series of queries and answers about animals. The
program consists of just seven lines (blank lines are ignored).
dog(fido).
dog(rover).
dog(henry).
cat(felix).
cat(michael).
cat(jane).
animal(X):-dog(X).
This program is not too hard to decipher. The first three lines are facts, with the
obvious interpretation that fido, rover and henry are all dogs. The next three facts
say that felix, michael and jane are all cats.
The final line is a rule saying that anything (let us call it X) is an animal if it is
a dog. Cat lovers may feel that cats can also claim to be called animals, but the
program is silent about this.
Having loaded the program, the user is then faced with the two character
symbol ?- which is called the system prompt. To check whether fido is a dog all
that is necessary is to type the query dog(fido) followed by a full stop and press the
'return' key, which indicates to the system that a response is needed. This gives the
complete dialogue:
?- dog(fido).
yes
Introduction
vii
The user can enter a series of queries at the prompt, asking for further
information.
?-dog(jane).
no
[Is jane a dog? No - a cat]
?- animal(fido).
yes
[Is fido an animal?]
[yes - because it is a dog and any dog is an
animal]
?- dog(X).
X = fido ;
X = rover ;
X = henry
[Is it possible to find anything, let us call it X, that
is a dog?]
?-animal(felix).
no
[felix is a cat and so does not qualify as an animal,
as far as the program is concerned]
[All 3 possible answers are provided]
Although straightforward, this example shows the two components of any
Prolog program, rules and facts, and also the use of queries that make Prolog
search through its facts and rules to work out the answer. Determining that fido is
an animal involves a very simple form of logical reasoning:
GIVEN THAT
any X is an animal if it is a dog
AND
fido is a dog
DEDUCE
fido must be an animal
This type of reasoning is fundamental to theorem proving in Mathematics and
to writing programs in Prolog.
Even very simple queries such as:
?-dog(fido).
can be looked at as asking the Prolog system to prove something, in this case that
fido is a dog. In the simplest cases it can do so simply by finding a fact such as
dog(fido) that it has been given. The system answers yes to indicate that this
simple 'theorem' has been proved.
You have now seen all three elements needed for logic programming in Prolog:
facts, rules and queries. There are no others. Everything else is built from them.
A word of warning is necessary at this stage. It is easy for the newcomer to get
started with Prolog, but do not be fooled by these examples into thinking that
Prolog is only capable of handling simple (Mickey Mouse?) problems. By putting
viii
Introduction
these very basic building blocks together Prolog provides a very powerful facility
for building programs to solve complex problems, especially ones involving
reasoning, but all Prolog programs are simple in form and are soundly based on the
mathematical idea of proving results from the facts and rules available.
Prolog has been used for a wide variety of applications. Many of these are in
Mathematics and Logic but many are not. Some examples of the second type of
application are
•
programs for processing a 'natural language' text, to answer questions
about its meaning, translate it to another language etc.
•
advisory systems for legal applications
•
applications for training
•
maintaining databases for the Human Genome project
•
a personnel system for a multi-national computer company
•
automatic story generation
•
analysing and measuring 'social networks'
•
a software engineering platform for supporting the development of
complex software systems
•
automatically generating legally correct licences and other documents in
multiple languages
•
an electronic support system for doctors.
Prolog is also being used as the basis for a standard 'knowledge representation
language' for the Semantic Web – the next generation of internet technology.
Prolog is one of the principal languages used by researchers in Artificial
Intelligence, with many applications developed in that field, especially in the form
of Expert Systems – programs that 'reason out' the solution to complex problems
using rules.
Many textbooks on Prolog assume that you are an experienced programmer
with a strong background in Mathematics, Logic or Artificial Intelligence
(preferably all three). This book makes no such assumptions. It starts from scratch
and aims to take you to a point where you can write quite powerful programs in the
language, often with considerably fewer lines of program 'code' than would be
needed in other languages.
You do not need to be an experienced programmer to learn Prolog. Some initial
familiarity with basic computing concepts such as program, variable, constant and
function would make it easier to achieve, but paradoxically too much experience of
writing programs in other languages may make the task harder – it may be
necessary to unlearn bad habits of thinking learnt elsewhere.
Introduction
ix
Some Technical Details
Experienced programmers will search this book in vain for such standard language
features as variable declarations, subroutines, methods, for loops, while loops or
assignment statements. (If you don't know what these terms mean, don't worry –
you will not be needing them.)
On the other hand experienced readers may like to know that Prolog has a
straightforward uniform syntax, programs that are equivalent to a database of facts
and rules, a built-in theorem prover with automatic backtracking, list processing,
recursion and facilities for modifying programs (or databases) at run-time. (You
probably will not know what most of these mean either – but you will be using all
of them by the end of this book.)
Prolog lends itself to a style of programming making particular use of two
powerful techniques: recursion and list processing. In many cases algorithms that
would require substantial amounts of coding in other languages can be
implemented in a few lines in Prolog.
There are many versions of Prolog available for PC, Macintosh and Unix
systems, including versions for Microsoft Windows, to link Prolog to an Oracle
relational database and for use with 'object-oriented' program design. These range
from commercial systems with many features to public domain and 'freeware'
versions. Some of these are listed (in alphabetical order) below, together with web
addresses at which more information can be found.
•
Amzi! Prolog
http://www.amzi.com/products/prolog_products.htm
•
B-Prolog
http://www.probp.com/
•
Ciao Prolog
http://clip.dia.fi.upm.es/Software/Ciao/
•
GNU Prolog
http://gnu-prolog.inria.fr/
•
Logic Programming Associates Prolog (versions for Windows, DOS and
Macintosh)
http://www.lpa.co.uk
•
Open Prolog (for Apple Macintosh)
http://www.cs.tcd.ie/open-prolog/
•
PD Prolog (a public domain version for MS-DOS only)
http://www-2.cs.cmu.edu/afs/cs/project/
ai-repository/ai/lang/prolog/impl/prolog/pdprolog/0.html
•
SICStus Prolog
http://www.sics.se/isl/sicstuswww/site/index.html
x
Introduction
•
SWI Prolog
http://www.swi-prolog.org/
•
Turbo Prolog (an old version that only runs in MS-DOS)
http://www.fraber.de/university/prolog/tprolog.html
•
Visual Prolog
http://www.visual-prolog.com/
•
W-Prolog (a Prolog-like language that runs in a web browser)
http://goanna.cs.rmit.edu.au/~winikoff/wp/
•
YAP Prolog
http://www.ncc.up.pt/~vsc/Yap/
The programs in this book are all written using the standard 'Edinburgh syntax'
and should run unchanged in virtually any version of Prolog you encounter
(unfortunately, finding occasional subtle differences between implementations is
one of the occupational hazards of learning any programming language). Features
such as graphical interfaces, links to external databases etc. have deliberately not
been included, as these generally vary from one implementation to another. All the
examples given have been tested using Win-Prolog version 4.110 from the British
company Logic Programming Associates (www.lpa.co.uk).
Each chapter has a number of self-assessment exercises to enable you to check
your progress. A full glossary of the technical terms used completes the book.
Contents
1. Getting Started
1
1.1 Starting Prolog
1.2 Prolog Programs
1.3 Data Objects in Prolog: Prolog Terms
Practical Exercise 1
1
3
8
12
2. Clauses and Predicates
13
2.1 Clauses
2.2 Predicates
2.3 Loading Clauses
2.4 Variables
Practical Exercise 2
13
15
19
21
27
3. Satisfying Goals
29
Introduction
3.1 Unification
3.1.1 Unifying Call Terms
3.2 Evaluating Goals
3.3 Backtracking
3.4 Satisfying Goals: A Summary
3.5 Removing Common Variables
3.6 A Note on Declarative Programming
Practical Exercise 3
29
31
32
35
39
50
52
52
55
4. Operators and Arithmetic
57
4.1 Operators
4.2 Arithmetic
4.3 Equality Operators
4.4 Logical Operators
Practical Exercise 4
57
59
63
66
68
xii
Contents
5. Input and Output
69
Introduction
5.1 Outputting Terms
5.2 Inputting Terms
5.3 Input and Output Using Characters
5.4 Outputting Characters
5.5 Inputting Characters
5.6 Using Characters: Examples
5.7 Input and Output Using Files
5.8 File Output: Changing the Current Output Stream
5.9 File Input: Changing the Current Input Stream
5.9.1 Reading from Files: End of File
5.9.2 Reading from Files: End of Record
5.10 Using Files: Examples
Practical Exercise 5
69
69
71
72
73
73
74
77
77
78
78
79
79
82
6. Loops
85
Introduction
6.1 Looping a Fixed Number of Times
6.2 Looping Until a Condition Is Satisfied
6.2.1 Recursion
6.2.2 Using the 'repeat' Predicate
6.3 Backtracking with Failure
6.3.1 Searching the Prolog Database
6.3.2 Finding Multiple Solutions
Practical Exercise 6
85
85
89
89
91
94
94
96
97
7. Preventing Backtracking
99
Introduction
7.1 The Cut Predicate
7.2 Cut with Failure
Practical Exercise 7
99
99
106
108
8. Changing the Prolog Database
109
8.1 Changing the Database: Adding and Deleting Clauses
8.2 Adding Clauses
8.3 Deleting Clauses
8.4 Changing the Database: Example
8.5 Maintaining a Database of Facts
Practical Exercise 8
109
110
111
112
114
118
Contents
9. List Processing
119
9.1 Representing Data as Lists
9.2 Notation for Lists
9.3 Decomposing a List
9.4 Built-in Predicate: member
9.5 Built-in Predicate: length
9.6 Built-in Predicate: reverse
9.7 Built-in Predicate: append
9.8 List Processing: Examples
9.9 Using findall/3 to Create a List
Practical Exercise 9
119
120
122
125
126
126
127
129
133
136
10. String Processing
137
10.1 Converting Strings of Characters To and From Lists
10.2 Joining Two Strings
10.3 Trimming a String
10.4 Inputting a String of Characters
10.5 Searching a String
10.6 Dividing a String into Its Component Parts
Practical Exercise 10
137
139
139
141
142
144
147
11. More Advanced Features
149
Introduction
11.1 Extending Prolog: Arithmetic
11.2 Extending Prolog: Operations on Strings
11.3 Extending Prolog: Sets
11.4 Processing Terms
Practical Exercise 11
149
149
155
157
160
166
Appendix 1. Built-in Predicates
167
Appendix 2. Built-in Operators
179
Appendix 3. Specimen Solutions to Practical Exercises
185
Appendix 4. Glossary
209
Index
221
xiii
1
Getting Started
Chapter Aims
After reading this chapter you should be able to:
•
Write and load a simple Prolog program
•
Enter goals at the Prolog system prompt
•
Understand the basic terminology of the Prolog language
•
Distinguish between different types of term (data objects).
1.1 Starting Prolog
Starting the Prolog system is usually straightforward, but the precise details will
vary from one version to another. Consult the documentation if necessary. Starting
Prolog will generally produce a number of lines of headings followed by a line
containing just
?This is the system prompt. (In some versions of Prolog a slightly different
combination of characters may be used.)
The prompt indicates that the Prolog system is ready for the user to enter a
sequence of one or more goals, which must be terminated by a full stop, for
example:
?- write('Hello World'),nl,write('Welcome to Prolog'),nl.
2
Logic Programming With Prolog
nl stands for 'start a new line', as will be explained later. Like all other user
input, the above line does not have any effect until the 'return' key is pressed.
Doing so produces the output
Hello World
Welcome to Prolog
yes
followed by a further system prompt ?-.
In this book a sequence of goals entered by the user will generally be shown
preceded by the ?- prompt. The prompt must not be typed by the user. It is
generated automatically by the Prolog system to show that it is ready to receive a
sequence of goals.
In the above example, the user has entered a sequence of four goals:
write('Hello World'), nl (twice) and write('Welcome to Prolog'). The commas
separating the goals signify 'and'.
In order for the sequence of goals
write('Hello World'),nl,write('Welcome to Prolog'),nl
to succeed each of the following goals has to succeed in order:
write('Hello World')
Hello World has to be displayed on the user's screen
nl
a new line has to be output to the user's screen
write('Welcome to Prolog')
Welcome to Prolog has to be displayed on the user's screen
nl
A new line has to be output to the user's screen.
The Prolog system can achieve all these goals simply by outputting lines of text
to the user's screen. It does so and then outputs yes to indicate that the sequence of
goals has succeeded.
From the system's point of view, the important issue is whether or not the
sequence of goals entered by the user succeeds. The generation of output to the
screen is considered much less important and is described as (merely) a side effect
of evaluating the goals write('Hello World') etc.
The meanings of write and nl are pre-defined by the Prolog system. They are
known as built-in predicates, sometimes abbreviated to BIPs.
Two other built-in predicates that are provided as standard in almost all
versions of Prolog are halt and statistics.
?-halt.
Getting Started
3
causes the Prolog system to terminate.
?- statistics.
causes system statistics (of value mainly to more experienced users) such as the
following to be generated.
Memory Statistics
-----------------------Reset Space
Heap Space
Input Space
Output Space
Free Bytes
-------------65536
261850
65536
65536
Total Bytes
--------------65536
262130
65536
65536
Total Elapsed Time 626.871
(100%)
Active Processing
Waiting for Input
yes
( 0%)
(100%)
2.704
623.977
Percent
---------(100%)
(100%)
(100%)
(100%)
Note that this output ends with the word yes, signifying that the goal has
succeeded, as statistics, halt and many other built-in predicates always do. Their
value lies in the side effects (generating statistics etc.) produced when they are
evaluated.
A sequence of one or more goals entered by the user at the prompt is often
called a query. We will generally use the term 'sequence of goals' in this book.
1.2 Prolog Programs
Entering a goal or a sequence of goals at the system prompt using only built-in
predicates would be of little value in itself. The normal way of working is for the
user to load a program written in the Prolog language and then enter a sequence of
one or more goals at the prompt, or possibly several sequences in succession, to
make use of the information that has been loaded into the database.
The simplest (and most usual) way to create a Prolog program is to type it into
a text editor and save it as a text file, say prog1.pl.
This is a simple example of a Prolog program. It has three components, known
as clauses, each terminated by a full stop. Note the use of blank lines to improve
readability – they are ignored.
dog(fido).
cat(felix).
animal(X):-dog(X).
4
Logic Programming With Prolog
The program can then be loaded for use by the Prolog system using the built-in
predicate consult.
?-consult('prog1.pl').
Provided that the file prog1.pl exists and the program is syntactically correct,
i.e. contains valid clauses, the goal will succeed and as a side effect produce one or
more lines of output to confirm that the program has been read correctly, e.g.
?# 0.00 seconds to consult prog1.pl
?If the Prolog system has a graphical user interface, there will probably be a 'Load'
option provided on a menu as an alternative to using consult. This and other menu
options such as 'exit' are not a standard part of the Prolog language and will not be
described in this book.
Loading a program simply causes the clauses to be placed in a storage area
called the Prolog database. Entering a sequence of one or more goals in response
to the system prompt causes Prolog to search for and use the clauses necessary to
evaluate the goal(s). Once placed in the database the clauses generally remain there
until the user exits from the Prolog system and so can be used to evaluate further
goals entered by the user.
Terminology
In the program above the three lines:
dog(fido).
cat(felix).
animal(X):-dog(X).
are all clauses. Each clause is terminated by a full stop. Apart from comments and
blank lines, Prolog programs consist only of a sequence of clauses. All clauses are
either facts or rules.
dog(fido) and cat(felix) are examples of facts. They can be interpreted in a
natural way as meaning 'fido is a dog' and 'felix is a cat'.
dog is called a predicate. It has one argument, the word fido enclosed in ( ).
fido is called an atom (meaning a constant which is not a number).
The final line of the program
animal(X):-dog(X).
Getting Started
5
is a rule. The :- character (colon and hyphen) can be read as 'if'. X is called a
variable. The meaning of a variable used in a rule or fact is described in Chapter 2.
In this context X represents any value, as long as it is the same value both times.
The rule can be read in a natural way as X is an animal if X is a dog (for any X).
From the above clauses it is simple (for humans) to deduce that fido is an
animal. Prolog can also make such deductions:
?- animal(fido).
yes
However there is no evidence to imply that felix is an animal:
?- animal(felix).
no
More Terminology
We say that a goal succeeds or fails, or alternatively that it is satisfied or cannot be
satisfied. The term evaluating a goal is used to mean determining whether or not it
is satisfied. Equivalently, we can say that a goal evaluates to true (i.e. succeeds) or
false (i.e. fails). This all fits in well with the everyday definition of a goal as
'something to be achieved'.
Note that sometimes a goal entered by the user can be interpreted as a
command, e.g.
?-halt.
'Exit from the Prolog system.'
At other times it can be regarded as a question, e.g.
?-animal(fido).
'Is fido an animal?'
Here is another program about animals. This one comprises eight clauses. All
text between /* and */ is taken to be a comment and ignored.
/* Animals Program 1
*/
dog(fido).
cat(mary). dog(rover).
dog(tom). cat(harry).
dog(henry).
cat(bill). cat(steve).
/* Apart from comments and blank lines, which are
ignored, Prolog programs consist of a number of
clauses. A clause is always terminated by a full stop.
6
Logic Programming With Prolog
It may run over more than one line, or there may be
several on the same line, separated by at least one
space. There are two types of clause: facts and rules.
dog(tom) is an example of a fact */
There are four clauses for predicate dog and four for predicate cat. We say that
the program comprises four clauses defining the dog predicate and four defining
the cat predicate.
Assuming that the program has been saved in a text file 'animals1.pl', the output
generated by loading the program and entering a sequence of goals at the system
prompt is given below.
?-consult('animals1.pl').
# 0.01 seconds to consult animals1.pl
System prompt
animals1.pl loaded using consult
?-dog(fido).
yes
?-dog(daisy).
No
?- dog(X).
X = fido
pauses – user presses return key
?- dog(Y).
Y = fido ;
Y = rover ;
Y = tom ;
Y = henry
pauses – user presses ;
pauses – user presses ;
pauses – user presses ;
No pause – goes on to next line
?- cat(X).
X = mary ;
X = harry
pauses – user presses ;
pauses – user presses return
?- listing(dog).
List all the clauses defining predicate dog
/* dog/1 */
dog( fido ).
dog( rover ).
dog( tom ).
dog( henry ).
yes
?-
Getting Started
7
There are several new features of Prolog introduced in this example. The query
?- dog(X).
(a single goal) means 'find a value of X for which the goal dog(X) is satisfied', or
effectively 'find a value of X which is the name of a dog'. Prolog answers
X=fido
However there are other possible answers (rover, tom and henry). Because of
this Prolog pauses and waits for the user to press the 'return' key before it outputs
the system prompt ?-.
The next query entered is
?- dog(Y).
This is essentially the same query as before. It is unimportant which variable (X
or Y) is used. The query means 'find a value of Y which is the name of a dog'.
Prolog answers
Y = fido
and again pauses. This time the user presses the ; (semicolon) key. Prolog now
looks for an alternative solution or, more precisely, an alternative value of Y that
satisfies the goal dog(Y). It replies
Y = rover
It pauses again and the user again presses the ; key. A further solution is given
Y = tom
Prolog pauses again. The user again presses the ; key, producing a further
solution
Y = henry
This time there are no more solutions available and Prolog recognises this by
not pausing, but immediately going on to output the system prompt ?-.
The process of finding alternative ways of satisfying a goal by entering a
semicolon at the system prompt is known as backtracking, or more precisely
'forcing the Prolog system to backtrack'. Backtracking will be discussed in more
detail in Chapter 3.
The example also introduces a new built-in predicate. Entering the goal
?-listing(dog).
8
Logic Programming With Prolog
causes Prolog to list all four clauses defining predicate dog, in the order in which
they were loaded into the database (which is the same as the order in which they
appeared in file animals1.pl).
The next example shows more about the use of variables in queries. The
sequence of goals
?-cat(X),dog(Y).
gives all possible combinations of a dog and a cat.
?-cat(X),dog(Y).
X = mary ,
Y = fido ;
X = mary ,
Y = rover ;
X = mary ,
Y = tom ;
X = mary ,
Y = henry ;
etc.
By contrast, the sequence of goals
?-cat(X),dog(X).
gives all animals which are both a cat and a dog (there are no such animals in the
database). Although X stands for 'any value' in both cat(X) and dog(X) they must
both be the same value.
?- cat(X),dog(X).
no
1.3 Data Objects in Prolog: Prolog Terms
The data objects in Prolog are called terms. Examples of terms that have been used
in Prolog programs so far in this book are fido, dog(henry), X and cat(X).
There are several different types of term, which are listed below.
Getting Started
9
(1) Numbers
All versions of Prolog allow the use of integers (whole numbers). They are written
as any sequence of numerals from 0 to 9, optionally preceded by a + or - sign, for
example:
623
-47
+5
025
Most versions of Prolog also allow the use of numbers with decimal points.
They are written in the same way as integers, but contain a single decimal point,
anywhere except before an optional + or - sign, e.g.
6.43
-.245
+256.
(2) Atoms
Atoms are constants that do not have numerical values. There are three ways in
which atoms can be written.
(a) Any sequence of one or more letters (upper or lower case), numerals and
underscores, beginning with a lower case letter, e.g.
john
today_is_Tuesday
fred_jones
a32_BCD
but not
Today
today-is-Tuesday
32abc
(b) Any sequence of characters enclosed in single quotes, including spaces and
upper case letters, e.g.
'Today is Tuesday'
'today-is-Tuesday'
'32abc'
(c) Any sequence of one or more special characters from a list that includes the
following + - * / > < = & # @ :
10
Logic Programming With Prolog
Examples
+++
>=
>
+-(3) Variables
In a query a variable is a name used to stand for a term that is to be determined,
e.g. variable X may stand for atom dog, the number 12.3, or a compound term or a
list (both to be described below). The meaning of a variable when used in a rule or
fact is described in Chapter 2.
The name of a variable is denoted by any sequence of one or more letters
(upper or lower case), numerals and underscores, beginning with an upper case
letter or underscore, e.g.
X
Author
Person_A
_123A
but not
45_ABC
Person-A
author
Note: The variable _ which consists of just a single underscore is known as the
anonymous variable and is reserved for a special purpose (see Chapter 2).
(4) Compound Terms
Compound terms are of fundamental importance in writing Prolog programs. A
compound term is a structured data type that begins with an atom, known here as a
functor. The functor is followed by a sequence of one or more arguments, which
are enclosed in brackets and separated by commas. The general form is
functor(t1,t2, … ,tn)
n ≥1
If you are familiar with other programming languages, you may find it helpful
to think of a compound term as representing a record structure. The functor
represents the name of the record, while the arguments represent the record fields.
The number of arguments a compound term has is called its arity. Some
examples of compound terms are:
likes(paul,prolog)
read(X)
Getting Started
11
dog(henry)
cat(X)
>(3,2)
person('john smith',32,doctor,london)
Each argument of a compound term must be a term, which can be of any kind
including a compound term. Thus some more complex examples of compound
terms are:
likes(dog(henry),Y)
pred3(alpha,beta,gamma,Q)
pred(A,B,likes(X,Y),-4,pred2(3,pred3(alpha,beta,gamma,Q)))
(5) Lists
A list is often considered to be a special type of compound term, but in this book it
will be treated as a separate type of data object.
Lists are written as an unlimited number of arguments (known as list elements)
enclosed in square brackets and separated by commas, e.g. [dog,cat,fish,man].
Unlike the arity of a compound term, the number of elements a list has does not
have to be decided in advance when a program is written, as will be explained in
Chapter 9. This can be extremely useful.
At this stage, all that it is necessary to know is that an element of a list may be a
term of any kind, including a compound term or another list, e.g.
[dog,cat,y,mypred(A,b,c),[p,q,R],z]
[[john,28],[mary,56,teacher],robert,parent(victoria,albert),[a,b,[c,d,e],f],29]
[[portsmouth,edinburgh,london,dover],[portsmouth,london,edinburgh],[glasgow]]
A list with no elements is known as the empty list. It is written as [].
(6) Other Types of Term
Some dialects of Prolog allow other types of term, e.g. character strings. These
will not be described in this book. However, it is possible to use atoms to perform a
rudimentary type of string processing (see Chapter 10).
Atoms and compound terms have a special importance in Prolog clauses and
are known collectively as call terms. We will return to this in future chapters.
Chapter Summary
This chapter shows how to write simple Prolog programs, load them into the
Prolog database and enter goals that can be evaluated using them. It also introduces
basic terminology and the different types of data object (terms).
12
Logic Programming With Prolog
Practical Exercise 1
Specimen solutions to all the Practical Exercises are given in Appendix 3.
(1) Create a disk file animals.pl containing Animals Program 1 (leaving out the
comments). Start up Prolog and load your program.
Test your program with the queries given in the text and some others of your own.
(2) Write a program to put facts indicating that a lion, a tiger and a cow are animals
into the database and to record that two of them (lion and tiger) are carnivores.
Save your program to a disk file and load it. Check that the database is correct
using listing.
Enter goals to test whether:
(a) there is such an animal as a tiger in the database
(b) a cow and a tiger are both in the database (a conjunction of two goals)
(c) a lion is an animal and also a carnivore
(d) a cow is an animal and also a carnivore.
(3) Try to predict what Prolog will output in response to each of the following
goals, and then try them.
?-write(hello).
?-write(Hello).
?-write('Hello!').
?-write('Hello!'),nl.
?-100=100.
?-100=1000/10.
?-100 is 1000/10.
?-1000 is 100*10.
?-2 is (5+7)/6.
?-74 is (5+7)*6.
2
Clauses and Predicates
Chapter Aims
After reading this chapter you should be able to:
•
Identify the components of rules and facts
•
Explain the meaning of the term predicate
•
Make correct use of variables in goals and clauses.
2.1 Clauses
Apart from comments and blank lines, which are ignored, a Prolog program
consists of a succession of clauses. A clause can run over more than one line or
there may be several on the same line. A clause is terminated by a dot character,
followed by at least one 'white space' character, e.g. a space or a carriage return.
There are two types of clause: facts and rules. Facts are of the form
head.
head is called the head of the clause. It takes the same form as a goal entered by
the user at the prompt, i.e. it must be an atom or a compound term. Atoms and
compound terms are known collectively as call terms. The significance of call
terms will be explained in Chapter 3.
14
Logic Programming With Prolog
Some examples of facts are:
christmas.
likes(john,mary).
likes(X,prolog).
dog(fido).
Rules are of the form:
head:-t1,t2, … , tk. (k>=1)
head is called the head of the clause (or the head of the rule) and, as for facts,
must be a call term, i.e. an atom or a compound term.
:- is called the neck of the clause (or the 'neck operator'). It is read as 'if'.
t1,t2, … , tk is called the body of the clause (or the body of the rule). It specifies
the conditions that must be met in order for the conclusion, represented by the
head, to be satisfied. The body consists of one or more components, separated by
commas. The components are goals and the commas are read as 'and'.
Each goal must be a call term, i.e. an atom or a compound term. A rule can be
read as 'head is true if t1, t2, …, tk are all true'.
The head of a rule can also be viewed as a goal with the components of its body
viewed as subgoals. Thus another reading of a rule is 'to achieve goal head, it is
necessary to achieve subgoals t1, t2,…, tk in turn'.
Some examples of rules are:
large_animal(X):-animal(X),large(X).
grandparent(X,Y):-father(X,Z),parent(Z,Y).
go:-write('hello world'),nl.
Here is another version of the animals program, which includes both facts and
rules.
/* Animals Program 2
*/
dog(fido). large(fido).
cat(mary). large(mary).
dog(rover). dog(jane).
dog(tom). large(tom). cat(harry).
Clauses and Predicates
15
dog(fred). dog(henry).
cat(bill). cat(steve).
small(henry). large(fred).
large(steve). large(jim).
large(mike).
large_animal(X):- dog(X),large(X).
large_animal(Z):- cat(Z),large(Z).
fido, mary, jane etc. are atoms, i.e. constants, indicated by their initial lower
case letters. X and Y are variables, indicated by their initial capital letters.
The first 18 clauses are facts. The final two clauses are rules.
2.2 Predicates
The following simple program has five clauses. For each of the first three clauses,
the head is a compound term with functor parent and arity 2 (i.e. two arguments).
parent(victoria,albert).
parent(X,Y):-father(X,Y).
parent(X,Y):-mother(X,Y).
father(john,henry).
mother(jane,henry).
It is possible (although likely to cause confusion) for the program also to
include clauses for which the head has functor parent, but a different arity, for
example
parent(john).
parent(X):-son(X,Y).
/* X is a parent if X has a son Y */
It is also possible for parent to be used as an atom in the same program, for
example in the fact
animal(parent).
but this too is likely to cause confusion.
16
Logic Programming With Prolog
All the clauses (facts and rules) for which the head has a given combination of
functor and arity comprise a definition of a predicate. The clauses do not have to
appear as consecutive lines of a program but it makes programs easier to read if
they do.
The clauses given above define two predicates with the name parent, one with
arity two and the other with arity one. These can be written (in textbooks, reference
manuals etc., not in programs) as parent/2 and parent/1, to distinguish between
them. When there is no risk of ambiguity, it is customary to refer to a predicate as
just dog, large_animal etc.
An atom appearing as a fact or as the head of a rule, e.g.
christmas.
go:-parent(john,B),
write('john has a child named '),
write(B),nl.
can be regarded as a predicate with no arguments, e.g. go/0.
There are five predicates defined in Animals Program 2: dog/1, cat/1, large/1,
small/1 and large_animal/1. The first 18 clauses are facts defining the predicates
dog/1, cat/1, large/1 and small/1 (6, 4, 7 and 1 clauses, respectively). The final
two clauses are rules, which together define the predicate large_animal/1.
Declarative and Procedural Interpretations of Rules
Rules have both a declarative and a procedural interpretation. For example, the
declarative interpretation of the rule
chases(X,Y):-dog(X),cat(Y),write(X),
write(' chases '),write(Y),nl.
is: 'chases(X,Y) is true if dog(X) is true and cat(Y) is true and write(X) is true,
etc.'
The procedural interpretation is 'To satisfy chases(X,Y), first satisfy dog(X),
then satisfy cat(Y), then satisfy write(X), etc.'
Facts are generally interpreted declaratively, e.g.
dog(fido).
is read as 'fido is a dog'.
The order of the clauses defining a predicate and the order of the goals in the
body of each rule are irrelevant to the declarative interpretation but of vital
importance to the procedural interpretation and thus to determining whether or not
the sequence of goals entered by the user at the system prompt is satisfied. When
Clauses and Predicates
17
evaluating a goal, the clauses in the database are examined from top to bottom.
Where necessary, the goals in the body of a rule are examined from left to right.
This topic will be discussed in detail in Chapter 3.
A user's program comprises facts and rules that define new predicates. These
are called user-defined predicates. In addition there are standard predicates predefined by the Prolog system. These are known as built-in predicates (BIPs) and
may not be redefined by a user program. Some examples are: write/1, nl/0,
repeat/0, member/2, append/3, consult/1, halt/0. Some BIPs are common to all
versions of Prolog. Others are version-dependent.
Two of the most commonly used built-in predicates are write/1 and nl/0.
The write/1 predicate takes a term as its argument, e.g.
write(hello)
write(X)
write('hello world')
Providing its argument is a valid term, the write predicate always succeeds and
as a side effect writes the value of the term to the user's screen. To be more precise
it is output to the current output stream, which by default will be assumed to be the
user's screen. Information about output to other devices is given in Chapter 5. If the
argument is a quoted atom, e.g. 'hello world', the quotes are not output.
The nl/0 predicate is an atom, i.e. a predicate that takes no arguments. The
predicate always succeeds and as a side effect starts a new line on the user's screen.
The name of a user-defined predicate (the functor) can be any atom, with a few
exceptions, except that you may not redefine any of the Prolog system's built-in
predicates. You are most unlikely to want to redefine the write/1 predicate by
putting a clause such as
write(27).
or
write(X):-dog(X).
in your programs, but if you do the system will give an error message such as
'illegal attempt to redefine a built-in predicate'.
The most important built-in predicates are described in Appendix 1. Each
version of Prolog is likely to have others – sometimes many others – and if you
accidentally use one of the same name and arity for one of your own predicates
you will get the 'illegal attempt to redefine a built-in predicate' error message,
which can be very puzzling.
It would be permitted to define a predicate with the same functor and a
different arity, e.g. write/3 but this is definitely best avoided.
18
Logic Programming With Prolog
Simplifying Entry of Goals
In developing or testing programs it can be tedious to enter repeatedly at the
system prompt a lengthy sequence of goals such as
?-dog(X),large(X),write(X),write(' is a large dog'),nl.
A commonly used programming technique is to define a predicate such as go/0
or start/0, with the above sequence of goals as the right-hand side of a rule, e.g.
go:-dog(X),large(X),write(X),
write(' is a large dog'),nl.
This enables goals entered at the prompt to be kept brief, e.g.
?-go.
Recursion
An important technique for defining predicates, which will be used frequently later
in this book, is to define them in terms of themselves. This is known as a recursive
definition. There are two forms of recursion.
(a) Direct recursion. Predicate pred1 is defined in terms of itself.
(b) Indirect recursion. Predicate pred1 is defined using pred2, which is defined
using pred3, …, which is defined using pred1.
The first form is more common. An example of it is
likes(john,X):-likes(X,Y),dog(Y).
which can be interpreted as 'john likes anyone who likes at least one dog'.
Predicates and Functions
The use of the term 'predicate' in Prolog is closely related to its use in mathematics.
Without going into technical details (this is not a book on mathematics) a predicate
can be thought of as a relationship between a number of values (its arguments)
such as likes(henry,mary) or X=Y, which can be either true or false.
This contrasts with a function, such as 6+4, the square root of 64 or the first
three characters of 'hello world', which can evaluate to a number, a string of
characters or some other value as well as true and false. Prolog does not make use
of functions except in arithmetic expressions (see Chapter 4).
Clauses and Predicates
19
2.3 Loading Clauses
There are two built-in predicates that can be used to load clauses into the Prolog
database: consult/1 and reconsult/1. Both will cause the clauses contained in a text
file to be loaded into the database as a side effect. However, there is a crucial
difference between them, which is illustrated by the following example. Supposing
file file1.pl contains
dog(fido).
dog(rover).
dog(jane).
dog(tom).
dog(fred).
cat(mary).
cat(harry).
small(henry).
large(fido).
large(mary).
large(tom).
large(fred).
large(steve).
large(jim).
and file file2.pl contains
dog(henry).
dog(fido).
cat(bill).
cat(steve).
large(mike).
large_animal(X):- dog(X),large(X).
large_animal(Z):- cat(Z),large(Z).
then entering the two goals
?-consult('file1.pl').
20
Logic Programming With Prolog
?-consult('file2.pl').
in succession at the prompt will put these clauses in the database.
dog(fido).
dog(rover).
dog(jane).
dog(tom).
dog(fred).
dog(henry).
dog(fido).
cat(mary).
cat(harry).
cat(bill).
cat(steve).
small(henry).
large(fido).
large(mary).
large(tom).
large(fred).
large(steve).
large(jim).
large(mike).
large_animal(X):- dog(X),large(X).
large_animal(Z):- cat(Z),large(Z).
Effectively, the clauses loaded from the second file are added to those already
loaded from the first file, predicate by predicate, after those already there. Note that
dog(fido) now appears in the database twice. There is nothing in the Prolog system
to prevent this.
By contrast, entering the two goals
?-consult('file1.pl').
?-reconsult('file2.pl').
in succession at the prompt will put these clauses in the database.
Clauses and Predicates
21
dog(henry).
dog(fido).
cat(bill).
cat(steve).
small(henry).
large(mike).
large_animal(X):- dog(X),large(X).
large_animal(Z):- cat(Z),large(Z).
This is most unlikely to be what is intended. The predicate definitions in file2.pl
completely replace any previous clauses for the same predicates in the database.
New predicates are loaded in the usual way. In the above example:
•
the definitions of dog/1, cat/1 and large/1 replace those already in the
database
•
the definition of small/1 in file1.pl remains in the database
•
the definition of large_animal/1 in file2.pl is placed in the database.
Although this example shows reconsult at its most unhelpful, in normal
program development reconsult is routinely used. Some program developers may
choose to load a large program in several parts (taking care that they have no
predicates in common) using several consult goals, but a far more common
method of program development is to load an entire program (set of clauses) as a
single file, test it, then make changes, save the changes in a new version of the file
with the same name and reload the clauses from the file. For this to work properly
it is imperative to ensure that the old versions of the clauses are deleted each time.
This can be achieved by using consult the first time and then reconsult each
subsequent time.
The predicates consult and reconsult are used so frequently that in many
versions of Prolog a simplified notation is available, with ['file1.pl'] standing for
consult('file1.pl') and [-'file1.pl'] standing for reconsult('file1.pl').
2.4 Variables
Variables can be used in the head or body of a clause and in goals entered at the
system prompt. However, their interpretation depends on where they are used.
Variables in Goals
Variables in goals can be interpreted as meaning 'find values of the variables that
make the goal satisfied'. For example, the goal
22
Logic Programming With Prolog
?-large_animal(A).
can be read as 'find a value of A such that large_animal(A) is satisfied'.
A third version of the Animals Program is given below (only the clauses
additional to those in Animals Program 2 in Section 2.1 are shown).
/* Animals Program 3
*/
/* As Animals Program 2 but with the additional rules
given below */
chases(X,Y):dog(X),cat(Y),
write(X),write(' chases '),write(Y),nl.
/* chases is a predicate with two arguments*/
go:-chases(A,B).
/* go is a predicate with no arguments */
A goal such as
?-chases(X,Y).
means find values of variables X and Y to satisfy chases(X,Y).
To do this, Prolog searches through all the clauses defining the predicate chases
(there is only one in this case) from top to bottom until a matching clause is found.
It then works through the goals in the body of that clause one by one, working
from left to right, attempting to satisfy each one in turn. This process is described
in more detail in Chapter 3.
The output produced by loading Animals Program 3 and entering some typical
goals at the prompt is as follows.
?-consult('animals3.pl').
# 0.01 seconds to consult animals3.pl
System prompt
animals3.pl loaded
?- chases(X,Y).
fido chases mary
X = fido ,
Y = mary ;
User backtracks to find first two
solutions only.
fido chases harry
X = fido ,
Y = harry
Note use of write and nl predicates
Clauses and Predicates
23
?-chases(D,henry).
no
Nothing chases henry
?-go.
fido chases mary
yes
Note that no variable values are output.
(All output is from the write and nl
predicates.) Because of this, the user has
no opportunity to backtrack.
It should be noted that there is nothing to prevent the same answer being
generated more than once by backtracking. For example if the program is
chases(fido,mary):-fchasesm.
chases(fido,john).
chases(fido,mary):-freallychasesm.
fchasesm.
freallychasesm.
The query ?-chases(fido,X) will produce two identical answers out of three by
backtracking.
?- chases(fido,X).
X = mary ;
X = john ;
X = mary
?Binding Variables
Initially all variables used in a clause are said to be unbound, meaning that they do
not have values. When the Prolog system evaluates a goal some variables may be
given values such as dog, -6.4 etc. This is known as binding the variables. A
variable that has been bound may become unbound again and possibly then bound
to a different value by the process of backtracking, which will be described in
Chapter 3.
Lexical Scope of Variables
In a clause such as
parent(X,Y):-father(X,Y).
the variables X and Y are entirely unrelated to any other variables with the same
name used elsewhere. All occurrences of variables X and Y in the clause can be
24
Logic Programming With Prolog
replaced consistently by any other variables, e.g. by First_person and
Second_person giving
parent(First_person,Second_person):father(First_person,Second_person).
This does not change the meaning of the clause (or the user's program) in any
way. This is often expressed by saying that the lexical scope of a variable is the
clause in which it appears.
Universally Quantified Variables
If a variable appears in the head of a rule or fact it is taken to indicate that the rule
or fact applies for all possible values of the variable. For example, the rule
large_animal(X):-dog(X),large(X).
can be read as 'for all values of X, X is a large animal if X is a dog and X is large'.
Variable X is said to be universally quantified.
Existentially Quantified Variables
Suppose now that the database contains the following clauses:
person(frances,wilson,female,28,architect).
person(fred,jones,male,62,doctor).
person(paul,smith,male,45,plumber).
person(martin,williams,male,23,chemist).
person(mary,jones,female,24,programmer).
person(martin,johnson,male,47,solicitor).
man(A):-person(A,B,male,C,D).
The first six clauses (all facts) comprise the definition of predicate person/5,
which has five arguments with obvious interpretations, i.e. the forename, surname,
sex, age and occupation of the person represented by the corresponding fact.
The last clause is a rule, defined using the person predicate, which also has a
natural interpretation, i.e. 'for all A, A is a man if A is a person whose sex is male'.
As explained previously, the variable A in the head of the clause (representing
forename in this case) stands for 'for all A' and is said to be universally quantified.
What about variables B, C and D? It would be a very bad idea for them to be
taken to mean 'for all values of B, C and D'. In order to show that, say, paul is a
man, there would then need to be person clauses with the forename paul for all
possible surnames, ages and occupations, which is clearly not a reasonable
Clauses and Predicates
25
requirement. A far more helpful interpretation would be to take variable B to mean
'for at least one value of B' and similarly for variables C and D.
This is the convention used by the Prolog system. Thus the final clause in the
database means 'for all A, A is a man if there a person with forename A, surname B,
sex male, age C and occupation D, for at least one value of B, C and D'.
By virtue of the third person clause, paul qualifies as a man, with values smith,
45 and plumber for variables B, C and D respectively.
?- man(paul).
yes
The key distinction between variable A and variables B, C and D in the
definition of predicate man is that B, C and D do not appear in the head of the
clause.
The convention used by Prolog is that if a variable, say Y, appears in the body
of a clause but not in its head it is taken to mean 'there is (or there exists) at least
one value of Y'. Such variables are said to be existentially quantified. Thus the rule
dogowner(X):-dog(Y),owns(X,Y).
can be interpreted as meaning 'for all values of X, X is a dog owner if there is some
Y such that Y is a dog and X owns Y'.
The Anonymous Variable
In order to find whether there is a clause corresponding to anyone called paul in
the database, it is only necessary to enter a goal such as:
?- person(paul,Surname,Sex,Age,Occupation).
at the prompt. Prolog replies as follows:
Surname = smith ,
Sex = male ,
Age = 45 ,
Occupation = plumber
In many cases it may be that knowing the values of some or all of the last four
variables is of no importance. If it is only important to establish whether there is
someone with forename paul in the database an easier way is to use the goal:
?- person(paul,_,_,_,_).
yes
The underscore character _ denotes a special variable, called the anonymous
variable. This is used when the user does not care about the value of the variable.
26
Logic Programming With Prolog
If only the surname of any people named paul is of interest, this can be found
by making the other three variables anonymous in a goal, e.g.
?- person(paul,Surname,_,_,_).
Surname = smith
Similarly, if only the ages of all the people named martin in the database are of
interest, it would be simplest to enter the goal:
?- person(martin,_,_,Age,_).
This will give two answers by backtracking.
Age = 23 ;
Age = 47
The three anonymous variables are not bound, i.e. given values, as would
normally be expected.
Note that there is no assumption that all the anonymous variables have the same
value (in the above examples they do not). Entering the alternative goal
?- person(martin,X,X,Age,X).
with variable X instead of underscore each time, would produce the answer
no
as there are no clauses with first argument martin where the second, third and fifth
arguments are identical.
Chapter Summary
This chapter introduces the two types of Prolog clause, namely facts and rules and
their components. It also introduces the concept of a predicate and describes
different features of variables.
Clauses and Predicates
27
Practical Exercise 2
(1) Type the following program into a file and load it into Prolog.
/* Animals Database */
animal(mammal,tiger,carnivore,stripes).
animal(mammal,hyena,carnivore,ugly).
animal(mammal,lion,carnivore,mane).
animal(mammal,zebra,herbivore,stripes).
animal(bird,eagle,carnivore,large).
animal(bird,sparrow,scavenger,small).
animal(reptile,snake,carnivore,long).
animal(reptile,lizard,scavenger,small).
Devise and test goals to find (a) all the mammals, (b) all the carnivores that are
mammals, (c) all the mammals with stripes, (d) whether there is a reptile that has a
mane.
(2) Type the following program into a file
/* Dating Agency Database */
person(bill,male).
person(george,male).
person(alfred,male).
person(carol,female).
person(margaret,female).
person(jane,female).
Extend the program with a rule that defines a predicate couple with two
arguments, the first being the name of a man and the second the name of a woman.
Load your revised program into Prolog and test it.
3
Satisfying Goals
Chapter Aims
After reading this chapter you should be able to:
•
Determine whether two call terms unify and thus whether a goal can be
matched with a clause in the database
•
Understand how Prolog uses unification and backtracking to evaluate a
sequence of goals entered by the user.
Introduction
We can now look more closely at how Prolog satisfies goals. A general
understanding of this is essential for any non-trivial use of the language. A good
understanding can often enable the user to write powerful programs in a very
compact way, frequently using just a few clauses.
The process begins when the user enters a sequence of goals at the system
prompt, for example
?- owns(X,Y),dog(Y),write(X),nl.
The Prolog system attempts to satisfy each goal in turn, working from left to
right. When the goal involves variables, e.g. owns(X,Y), this generally involves
binding them to values, e.g. X to john and Y to fido. If all the goals succeed in turn,
the whole sequence of goals succeeds. The system will output the values of all the
variables that are used in the sequence of goals and any other text output as a side
effect by goals such as write(X) and nl.
30
Logic Programming With Prolog
?- owns(X,Y),dog(Y),write(X),nl.
john
X = john ,
Y = fido
If it is not possible to satisfy all the goals (simultaneously), the sequence of
goals will fail.
?- owns(X,Y),dog(Y),write(X),nl.
no
We will defer until Section 3.3 the issue of precisely what Prolog does if, say,
the first goal succeeds and the second fails.
Call Terms
Every goal must be a Prolog term, as defined in Chapter 1, but not any kind of
term. It may only be an atom or a compound term, not a number, variable, list or
any other type of term provided by some particular implementation of Prolog. This
restricted type of term is called a call term. Heads of clauses and goals in the
bodies of rules must also be call terms. The need for all three to take the same
(restricted) form is essential for what follows.
Every goal such as write('Hello World'), nl, dog(X) and go has a
corresponding predicate, in this case write/1, nl/0, dog/1 and go/0 respectively.
The name of the predicate (write, nl etc.) is called the functor. The number of
arguments it has is called the arity.
Goals relating to built-in predicates are evaluated in a way pre-defined by the
Prolog system, as was discussed for write/1 and nl/0 in Chapter 2. Goals relating
to user-defined predicates are evaluated by examining the database of rules and
facts loaded by the user.
Prolog attempts to satisfy a goal by matching it with the heads of clauses in the
database, working from top to bottom.
For example, the goal
?-dog(X).
might be matched with the fact
dog(fido).
to give the output
X=fido
A fundamental principle of evaluating user-defined goals in Prolog is that any
goal that cannot be satisfied using the facts and rules in the database fails. There is
no intermediate position, such as 'unknown' or 'not proven'. This is equivalent to
Satisfying Goals
31
making a very strong assumption about the database called the closed world
assumption: any conclusion that cannot be proved to follow from the facts and
rules in the database is false. There is no other information.
3.1 Unification
Given a goal to evaluate, Prolog works through the clauses in the database trying to
match the goal with each clause in turn, working from top to bottom until a match
is found. If no match is found the goal fails. The action taken if a match is found is
described in Section 3.2.
Prolog uses a very general form of matching known as unification, which
generally involves one or more variables being given values in order to make two
call terms identical. This is known as binding the variables to values. For example,
the terms dog(X) and dog(fido) can be unified by binding variable X to atom fido,
i.e. giving X the value fido. The terms owns(john,fido) and owns(P,Q) can be
unified by binding variables P and Q to atoms john and fido, respectively.
Initially all variables are unbound, i.e. do not have any value. Unlike for most
other programming languages, once a variable has been bound it can be made
unbound again and then perhaps be bound to a new value by backtracking, which
will be explained in Section 3.3.
The process of unifying a goal with the head of a clause is explained first. After
that unification will be used to explain how Prolog satisfies goals.
Warning: A Note on Terminology
The words unified, unify etc. are used in two different ways, which can sometimes
cause confusion.
When we say that 'two call terms are unified' we strictly mean that an attempt is
made to make the call terms identical (which generally involves binding variables
to values). This attempt may succeed or fail.
For example, the call terms likes(X,mary) and likes(john,Y) can be made
identical by binding variable X to atom john and variable Y to atom mary. In this
case we say that the unification succeeds. However there is no way of binding
variables to values that will make the call terms likes(X,mary) and dog(Z)
identical. In this case we say that the unification fails or that the call terms fail to
unify.
Expressions such as 'the unification of the two call terms succeeds' are often
abbreviated to just 'the two call terms are unified' or 'the two call terms unify'. The
intended meaning (the attempt or the successful attempt) is usually obvious from
the context, but it is a potential trap for the inexperienced!
32
Logic Programming With Prolog
3.1.1 Unifying Call Terms
The process is summarised in the following flowchart.
Are call terms both
constants?
Yes
No
Are call terms both
compound terms?
Succeeds if they are
the same constant,
otherwise fails.
No
Yes
Same functor and
arity?
No
Yes
Do arguments unify
pairwise?
No
Fails
Yes
Succeeds
Figure 3.1 Unifying Two Call Terms
There are three cases to consider. The simplest is when an atom is unified with
another atom. This succeeds if and only if the two atoms are the same, so
•
unifying atoms fido and fido succeeds
•
unifying atoms fido and 'fido' also succeeds, as the surrounding quotes are
not considered part of the atom itself
•
unifying atoms fido and rover fails.
A second possibility is that an atom is unified with a compound term, e.g. fido with
likes(john,mary). This always fails.
The third and by far the most common case is that two compound terms are
unified, e.g. likes(X,Y) with likes(john,mary) or dog(X) with likes(john,Y).
Unification fails unless the two compound terms have the same functor and the
same arity, i.e. the predicate is the same, so unifying dog(X) and likes(john,Y)
inevitably fails.
Satisfying Goals
33
Unifying two compound terms with the same functor and arity, e.g. the goal
person(X,Y,Z) with the head person(john,smith,27), requires the arguments of
the head and clause to be unified 'pairwise', working from left to right, i.e. the first
arguments of the two compound terms are unified, then their second arguments are
unified, and so on. So X is unified with john, then Y with smith, then Z with 27. If
all the pairs of arguments can be unified (as they can in this case) the unification of
the two compound terms succeeds. If not, it fails.
The arguments of a compound term can be terms of any kind, i.e. numbers,
variables and lists as well as atoms and compound terms. Unifying two terms of
this unrestricted kind involves considering more possibilities than unifying two call
terms.
•
Two atoms unify if and only if they are the same.
•
Two compound terms unify if and only if they have the same functor
and the same arity (i.e. the predicate is the same) and their arguments
can be unified pairwise, working from left to right.
•
Two numbers unify if and only if they are the same, so 7 unifies with
7, but not with 6.9.
•
Two unbound variables, say X and Y always unify, with the two
variables bound to each other.
•
An unbound variable and a term that is not a variable always unify,
with the variable bound to the term.
X and fido unify, with variable X bound to the atom fido
X and [a,b,c] unify, with X bound to list [a,b,c]
X and mypred(a,b,P,Q,R) unify, with X bound to mypred(a,b,P,Q,R)
•
A bound variable is treated as the value to which it is bound.
•
Two lists unify if and only if they have the same number of elements
and their elements can be unified pairwise, working from left to right.
[a,b,c] can be unified with [X,Y,c], with X bound to a and Y bound to b
[a,b,c] cannot be unified with [a,b,d]
•
[a,mypred(X,Y),K] can be unified with [P,Z,third], with variables P,
Z and K bound to atom a, compound term mypred(X,Y) and atom
third, respectively.
•
All other combinations of terms fail to unify.
Figure 3.2 Unifying Two Terms
Unification is probably easiest to understand if illustrated visually, to show the
related pairs of arguments. Some typical unifications are shown below.
34
Logic Programming With Prolog
person(X,Y,Z)
person(john,smith,27)
Succeeds with variables X, Y and Z bound to john, smith and 27,
respectively.
person(john,Y,23)
person(X,smith,27)
Fails because 23 cannot be unified with 27
pred1(X,Y,[a,b,c])
pred1(A,prolog,B)
Succeeds with variables X and A bound to each other, Y bound to atom
prolog and B bound to list [a,b,c].
Repeated Variables
A slightly more complicated case arises when a variable appears more than once in
a compound term.
pred2(X,X,man)
pred2(london,dog,A)
?
Here the first arguments of the two compound terms are unified successfully,
with X bound to the atom london. All other values of X in the first compound term
are also bound to the atom london and so are effectively replaced by that value
before any subsequent unification takes place. When Prolog comes to examine the
two second arguments, they are no longer X and dog but london and dog. These
are different atoms and so fail to unify.
pred2(X,X,man)
pred2(london,dog,A)
Fails because X cannot unify with both the atoms london and dog.
In general, after any pair of arguments are unified, all bound variables are
replaced by their values.
The next example shows a successful unification involving repeated variables.
Satisfying Goals
35
pred3(X,X,man)
pred3(london,london,A)
Succeeds with variables X and A bound to atoms london and man,
respectively.
This example shows a repeated variable in one of the arguments of a compound
term.
pred(alpha,beta,mypred(X,X,Y))
pred(P,Q,mypred(no,yes,maybe))
Fails.
P successfully unifies with alpha. Next Q unifies with beta. Then Prolog
attempts to unify the two third arguments, i.e. mypred(X,X,Y) and
mypred(no,yes,maybe). The first step is to unify variable X with the atom no.
This succeeds with X bound to no. Next the two second arguments are compared.
As X is bound to no, instead of X and yes the second arguments are now no and
yes, so the unification fails.
In this example, the second mypred argument is now no rather than yes, so
unification succeeds.
pred(alpha,beta,mypred(X,X,Y))
pred(P,Q,mypred(no,no,maybe))
Succeeds with variables P, Q, X and Y bound to atoms alpha, beta, no and
maybe, respectively.
3.2 Evaluating Goals
Given a goal such as go or dog(X) Prolog searches through the database from top
to bottom examining those clauses that have heads with the same functor and arity
until it finds the first one for which the head unifies with the goal. If there are none
the goal fails. If it does make a successful unification, the outcome depends on
whether the clause is a rule or a fact.
If the clause is a fact the goal succeeds immediately. If it is a rule, Prolog
evaluates the goals in the body of the rule one by one, from left to right. If they all
succeed, the original goal succeeds. (The case where they do not all succeed will
be covered in Section 3.3.)
We will use the phrase 'a goal matches a clause' to mean that it unifies with the
head of the clause.
36
Logic Programming With Prolog
Example
In this example, the goal is
?-pred(london,A).
It is assumed that the first clause in the database with predicate pred/2 and a
head that unifies with this goal is the following rule, which we will call Rule 1 for
ease of reference.
pred(X,'european capital'):capital(X,Y),european(Y),write(X),nl.
The unification binds X to the atom london and A to the atom 'european
capital'. The binding of X to london affects all occurrences of X in the rule. We
can show this diagrammatically as:
?-pred(london,A).
pred(london,'european capital'):capital(london,Y),european(Y),write(london),nl.
X is bound to london, A is bound to 'european capital'.
Next Prolog examines the goals in the body of Rule 1 one by one, working
from left to right. All of them have to be satisfied in order for the original goal to
succeed.
Evaluating each of these goals is carried out in precisely the same way as
evaluating the user's original goal. If a goal unifies with the head of a rule, this will
involve evaluation of the goals in the body of that rule, and so on.
We will assume that the first clause matched by goal capital(london,Y) is the
fact capital(london,england). The first goal in the body of Rule 1 is thus satisfied,
with Y bound to the atom england. This binding affects all occurrences of Y in the
body of Rule 1, not just the first one, so we now have
?-pred(london,A).
pred(london,'european capital'):capital(london,england),european(england),write(london),nl.
capital(london,england).
X is bound to london. A is bound to 'european capital'. Y is bound to
england.
It is now necessary to try to satisfy the second goal in the body of Rule 1,
which in rewritten form is european(england).
Satisfying Goals
37
This time we shall assume that the first clause in the database that has a head
that unifies with the goal is the rule
european(england):-write('God Save the Queen!'),nl.
We will call this Rule 2.
Prolog now tries to satisfy the goals in the body of Rule 2: write('God Save
the Queen!') and nl. It does this successfully, in the process outputting the line of
text
God Save the Queen!
as a side effect.
The first two goals in the body of Rule 1 have now been satisfied. There are
two more goals, which in rewritten form are write(london) and nl. Both of these
succeed, in the process outputting the line of text
london
as a side effect.
All the goals in the body of Rule 1 have now succeeded, so the goal that forms
its head succeeds, i.e. pred(london,'european capital').
This in turn means that the original goal entered by the user
?-pred(london,A).
succeeds, with A bound to 'european capital'.
The output produced by the Prolog system would be:
?- pred(london,A).
God Save the Queen!
london
A = 'european capital'
We can now see why output from write/1 and nl/0 goals is referred to by the
slightly dismissive term 'side effect'. The principal focus of the Prolog system is
the evaluation of goals (either entered by the user or in the bodies of rules), by
unification with the heads of clauses. Everything else is incidental. Of course, it is
frequently the side effects that are of most interest to the user.
This process of satisfying the user's goal creates linkages between the goal, the
heads of clauses and the goals in the bodies of rules. Although the process is
lengthy to describe, it is usually quite easy to visualise the linkages.
38
Logic Programming With Prolog
?-pred(london,A).
pred(london,'european capital'):capital(london,england),european(england),write(london),nl.
capital(london,england).
european(england):-write('God Save the Queen!'),nl.
X is bound to london. A is bound to 'european capital'. Y is bound to
england.
Note that the user's goal
?-pred(london,A).
has been placed to the right in the above diagram. That is because it has much in
common with a goal in the body of a rule. A sequence of goals entered by the user
at the prompt, for example
?- owns(X,Y),dog(Y),write(X),nl.
is treated in the same way as a sequence of goals in the body of an imaginary rule,
say succeed:-owns(X,Y),dog(Y),write(X),nl.
The process of evaluating a goal is summarised (in much simplified form) in
the following flowcharts. Note that the flowchart for evaluating a sequence of
goals refers to the one for evaluating a (single) goal, and vice versa.
Are there
more goals?
No
Yes
Succeeds
Evaluate next goal
Fails
See Section 3.3
Figure 3.2 Evaluating a Sequence of Goals
Sequence of goals
succeeds
Satisfying Goals
Is predicate a BIP?
Yes
No
Are there more clauses
in the database?
39
Goal is evaluated as
predefined by system
(may succeed or fail)
No
Goal fails
Yes
No
Can the goal be unified
with head of next clause?
Yes
Fails
Evaluate body of clause
(a sequence of goals)
Succeed
Goal succeeds
Figure 3.3 Evaluating a Goal
The principal issue that has been left unconsidered in this account is what
happens if evaluation of any of the goals fails. If it does, the Prolog system tries to
find another way of satisfying the most recently satisfied previous goal. This is
known as backtracking and is the topic of the next section. Unification and
backtracking together comprise the mechanism that Prolog uses to evaluate all
goals, whether entered by the user at the prompt or in the body of a rule.
3.3 Backtracking
Backtracking is the process of going back to a previous goal and trying to resatisfy
it, i.e. to find another way of satisfying it.
This section gives two very detailed accounts of the way that Prolog tries to
satisfy a sequence of goals using unification and backtracking. With practice it is
quite easy to work out the sequence of operations by visual inspection of the
database. However, it may be helpful to have a detailed account available for
reference.
40
Logic Programming With Prolog
The Family Relationships Example
This example is concerned with family relationships amongst a group of people.
The clauses shown below comprise 10 facts defining the mother/2 predicate, 9
facts defining the father/2 predicate and 6 clauses defining the parent/2 predicate.
[M1]
mother(ann,henry).
[M2]
mother(ann,mary).
[M3]
mother(jane,mark).
[M4]
mother(jane,francis).
[M5]
mother(annette,jonathan).
[M6]
mother(mary,bill).
[M7]
mother(janice,louise).
[M8]
mother(lucy,janet).
[M9]
mother(louise,caroline).
[M10]
mother(louise,martin).
[F1]
father(henry,jonathan).
[F2]
father(john,mary).
[F3]
father(francis,william).
[F4]
father(francis,louise).
[F5]
father(john,mark).
[F6]
father(gavin,lucy).
[F7]
father(john,francis).
[F8]
father(martin,david).
[F9]
father(martin,janet).
[P1]
parent(victoria,george).
[P2]
parent(victoria,edward).
[P3]
parent(X,Y):-write('mother?'),nl,mother(X,Y),
write('mother!'),nl.
[P4]
parent(A,B):-write('father?'),nl,father(A,B),
write('father!'),nl..
[P5]
parent(elizabeth,charles).
[P6]
parent(elizabeth,andrew).
Satisfying Goals
41
Facts such as
mother(jane,mark).
father(john,mark).
can be interpreted as meaning 'jane is the mother of mark' and 'john is the father of
mark', respectively.
Note that labels such as [M1] have been added here for reference purposes
only. They are not part of the clauses and must not be included in any program
files.
The facts relevant to the following examples can be shown diagrammatically as
follows (with 'f' standing for 'father').
john
f
f
f
mark
mary
francis
Example 1
Given the query
?-parent(john,Child),write('The child is '),write(Child),nl.
Prolog attempts to satisfy all the goals in the sequence (simultaneously) and in
doing so will find one or more possible values for variable Child. It starts with the
first goal parent(john,Child) and attempts to unify it with the head of each of the
clauses defining the predicate parent/2 in turn, working from top to bottom. It first
comes to clauses [P1] and [P2] but fails to match the goal with (i.e. unify the goal
with the head of) either of them. It next comes to clause [P3] and this time the goal
is successfully unified with the head of the clause, with X bound to john and
variables Y and Child bound to each other.
?-parent(john,Child),write('The child is '),write(Child),nl.
[P3] parent(john,Y):-write('mother?'),nl,mother(john,Y),write('mother!'),nl.
X is bound to john. Variables Y and Child are bound to each other.
The system now works through the goals in the body of rule [P3] trying to
make each one succeed in turn. It successfully evaluates the goals
write('mother?') and nl, outputting the line of text
42
Logic Programming With Prolog
mother?
as a side effect.
It then comes to the third of the goals, i.e. mother(john,Y). This does not unify
with the head of any of the clauses [M1] to [M10] which define the mother/2
predicate, so the goal fails.
The system now backtracks. It goes back to the most recently satisfied goal in
the body of [P3], moving from right to left, which is nl, and tries to resatisfy it, i.e.
to find another way of satisfying it.
Like many (but not all) built-in predicates, nl/0 is unresatisfiable, meaning that
it always fails when evaluated during backtracking.
Prolog now moves one further position to the left in the body of [P3], to the
goal write('mother?'). The predicate write/1 is also unresatisfiable, so this goal
also fails.
There are no further goals in the body of rule [P3], working from right to left,
so the system rejects rule [P3]. We now have simply
?-parent(john,Child),write('The child is '),write(Child),nl.
Variable Child is unbound.
with variable Child unbound.
Prolog now goes back to the most recently evaluated previous goal, which in
this case is parent(john,Child), and tries to resatisfy it. It continues searching
through the database for clauses defining the parent/2 predicate from the point it
had previously reached, i.e. clause [P3]. It first examines clause [P4] and
successfully unifies the goal with its head, with variable A bound to john and
variables B and Child bound to each other.
?-parent(john,Child),write('The child is '),write(Child),nl.
[P4] parent(john,B):-write('father?'),nl,father(john,B),write('father!'),nl.
A is bound to john. Variables B and Child are bound to each other.
The system now works through the goals in the body of the rule [P4] trying to
make each succeed in turn. The first two goals succeed, with the line of text
father?
output as a side effect.
The system now tries to satisfy the third goal, i.e. father(john,B). It searches
through the clauses defining the father/2 predicate in turn, from top to bottom.
The first clause it matches is [F2], which is a fact. This causes variable B to be
bound to atom mary. This in turn causes variable Child (which is bound to
variable B) to be bound to atom mary.
Satisfying Goals
43
?-parent(john,Child),write('The child is '),write(Child),nl.
[P4] parent(john,mary):write('father?'),nl,father(john,mary),write('father!'),nl.
[F2] father(john,mary).
A is bound to john. Variables B and Child are bound to each other and to
atom mary.
There are two further goals in the body of rule [P4], i.e. write('father!') and nl.
These both succeed with the line of text
father!
output as a side effect. All the goals in the body of [P4] have now succeeded, so
the head of the clause, which in rewritten form is parent(john,mary), succeeds.
The goal parent(john,Child) in the user's query therefore succeeds.
The first of the goals in the sequence entered by the user has now been
satisfied. There are three more goals in the sequence: write('The child is '),
write(Child) and nl. They all succeed, as a side effect outputting the line of text
The child is mary
All the goals in the user's query have now been satisfied. The Prolog system
outputs the value of all the variables used in the query. In this case, the only one is
Child.
?- parent(john,Child),write('The child is '),write(Child),nl.
mother?
father?
father!
The child is mary
Child = mary
Forcing the System to Backtrack to Find Further Solutions
The user can now force the system to backtrack to find a further solution or
solutions by entering a semicolon character. This works by forcing the most
recently satisfied goal, i.e. nl (the last goal in the user's query) to fail. The system
now backtracks to the previous goal in the sequence, i.e. write(Child). This too
fails on backtracking, as does the previous goal, i.e. write('The child is '). The
system backtracks a further step, to the first goal in the query, which is
parent(john,Child).
44
Logic Programming With Prolog
?-parent(john,Child),write('The child is '),write(Child),nl.
[P4] parent(john,mary):write('father?'),nl,father(john,mary),write('father!'),nl.
[F2] father(john,mary).
A is bound to john. Variables B and Child are bound to each other and to
atom mary.
The system attempts to find another way of satisfying it, beginning by trying to
find another way of satisfying the last goal in the body of [P4]. This is nl, which
fails on backtracking. So too does the previous goal write('father!').
It now attempts to resatisfy the previous goal in the body of [P4], working from
right to left, which is father(john,B). This process begins by rejecting the
unification with the head of [F2]. Prolog now continues to search through the
clauses defining the father/2 predicate for further unifications. The next successful
unification is with the head of clause [F5]. The terms father(john,B) and
father(john,mark) are unified with variable B bound to mark. This causes
variable Child also to be bound to mark.
?-parent(john,Child),write('The child is '),write(Child),nl.
[P4] parent(john,mark):write('father?'),nl,father(john,mark),write('father!'),nl.
[F5] father(john,mark).
A is bound to john. Variables B and Child are bound to each other and to
atom mark.
This gives a second solution to the user's goal, i.e. a second way of satisfying it.
Further backtracking will find a third solution, using clause [F7].
?-parent(john,Child),write('The child is '),write(Child),nl.
[P4] parent(john,francis):write('father?'),nl,father(john,francis),write('father!'),nl.
[F7] father(john,francis).
A is bound to john. Variables B and Child are bound to each other and to
atom francis.
?- parent(john,Child),write('The child is '),write(Child),nl.
mother?
father?
Satisfying Goals
45
father!
The child is mary
Child = mary ;
father!
The child is mark
Child = mark ;
father!
The child is francis
Child = francis
If the user again enters a semicolon to force the system to backtrack, the system
will again go through the backtracking sequence described above, until it reaches
the stage of attempting to resatisfy father(john,B), by rejecting the unification
with the head of clause [F7] previously found and continuing to search through the
clauses defining the father/2 predicate for further matches. As no further
unifications are found, the goal father(john,B) in the body of rule [P4] will now
fail.
The system now attempts to resatisfy the goal to the left of it in the body of rule
[P4]. This is nl, which always fails on backtracking. The next goal, again moving
to the left, is write('father?'), which also fails. There are no further goals in the
body of [P4], moving from right to left, so the system rejects rule [P4]. This brings
it back to the original goal parent(john,Child), which it tries to resatisfy. It
continues to search through the clauses defining the parent/2 predicate from the
point it previously reached ([P4]), but finds no further matches, so the goal fails.
As this is the first in the sequence of goals entered by the user, no further
backtracking is possible and the user's query finally fails.
?- parent(john,Child),write('The child is '),write(Child),nl.
mother?
father?
father!
The child is mary
Child = mary ;
father!
The child is mark
Child = mark ;
father!
The child is francis
Child = francis
?The system prompt is displayed to indicate that there are no more solutions
available by backtracking.
46
Logic Programming With Prolog
Example 2
In the following example the clauses in the database are as before, with the
addition of the clauses
[R1]
rich(jane).
[R2]
rich(john).
[R3]
rich(gavin).
[RF1]
rich_father(X,Y):-rich(X),father(X,Y).
Labels such as [R1] have again been added for ease of reference. They are not
part of the clauses.
Given the goal
?-rich_father(A,B).
Prolog starts by trying to unify the goal with the heads of all the clauses
defining the rich_father/2 predicate. There is only one, i.e. clause [RF1].
Unification succeeds and variables A and X are bound to each other. Variables B
and Y are also bound to each other.
?-rich_father(A,B).
[RF1] rich_father(X,Y):-rich(X),father(X,Y).
Variables A and X are bound to each other. Variables B and Y are bound to
each other.
Next Prolog tries to find a value of A satisfying the first goal in the body of rule
[RF1]. It does this by searching through the clauses defining the rich/1 predicate.
The first unification it finds is with the head of [R1], i.e. rich(jane). X is bound to
jane.
?-rich_father(A,B).
[RF1] rich_father(jane,Y):-rich(jane),father(jane,Y).
[R1] rich(jane).
Variables A and X are bound to each other and to atom jane. Variables B
and Y are bound to each other.
The system now tries to satisfy the goal father(jane,Y) by examining the
clauses defining the father/2 predicate, i.e. [F1] to [F9]. None of them unify with
the goal, so the system backtracks and attempts to resatisfy (i.e. find another
Satisfying Goals
47
solution to) the most recently satisfied goal, which is rich(X). It continues
searching through the clauses defining the rich/1 predicate, the next unification
found being with rich(john) (clause [R2]). Now X is bound to john, which in turn
causes A to be bound to john.
?-rich_father(A,B).
[RF1] rich_father(john,Y):-rich(john),father(john,Y).
[R2] rich(john).
Variables A and X are bound to each other and to atom john. Variables B
and Y are bound to each other.
The system now tries to satisfy the goal father(john,Y) by examining the
clauses defining the father/2 predicate, i.e. [F1] to [F9]. The first unification found
is with [F2], i.e. father(john,mary). Y is bound to mary.
?-rich_father(A,B).
[RF1] rich_father(john,mary):-rich(john),father(john,mary).
[R2] rich(john).
[F2] father(john,mary).
Variables A and X are bound to each other and to atom john. Variables B
and Y are bound to each other and to atom mary.
There are no more goals in the body of [RF1], so the rule succeeds. This in turn
causes the goal rich_father(A,B) to succeed, with A and B bound to john and
mary, respectively.
?- rich_father(A,B).
A = john ,
B = mary
The user can now force the system to backtrack to find further solutions by
entering a semicolon character. If so, it attempts to resatisfy the most recently
matched goal, i.e. father(john,Y) by rejecting the match with [F2] previously
found. This causes B and Y no longer to be bound to mary (they are still bound to
each other).
The system continues to search the clauses defining the father/2 predicate for
further matches. The next unification found is with the head of clause [F5].
Variable Y is bound to mark.
48
Logic Programming With Prolog
?-rich_father(A,B).
[RF1] rich_father(john,mark):-rich(john),father(john,mark).
[R2] rich(john).
[F5] father(john,mark).
Variables A and X are bound to each other and to atom john. Variables B
and Y are bound to each other and to atom mark.
This gives a second solution to the user's query. If the user forces the system to
backtrack again, it will find a third solution using clause [F7] father(john,francis).
?-rich_father(A,B).
[RF1] rich_father(john,francis):-rich(john),father(john,francis).
[R2] rich(john).
[F7] father(john,francis).
Variables A and X are bound to each other and to atom john. Variables B
and Y are bound to each other and to atom francis.
If the user forces the system to backtrack again, it will start by deeming that the
most recently satisfied goal, i.e. father(john,Y) has failed. This causes B and Y no
longer to be bound to francis (they are still bound to each other).
?-rich_father(A,B).
[RF1] rich_father(john,Y):-rich(john),father(john,Y).
[R2] rich(john).
Variables A and X are bound to each other and to atom john. Variables B
and Y are bound to each other.
The system will fail to find any further matches for the goal father(john,Y). It
will next attempt to find further solutions to the most recently satisfied previous
goal in [RF1], working from right to left. This is rich(X). This will succeed with X
now bound to gavin (clause [R3]).
Satisfying Goals
49
?-rich_father(A,B).
[RF1] rich_father(gavin,Y):-rich(gavin),father(gavin,Y).
[R3] rich(gavin).
Variables A and X are bound to each other and to atom gavin. Variables B
and Y are bound to each other.
Working from left to right again, the system will now try to satisfy the goal
father(gavin,Y). This will unify with the head of just one of the father/2 clauses,
namely with clause [F6] father(gavin,lucy), with variable Y bound to lucy.
?-rich_father(A,B).
[RF1] rich_father(gavin,lucy):-rich(gavin),father(gavin,lucy).
[R3] rich(gavin).
[F6] father(gavin,lucy).
Variables A and X are bound to each other and to atom gavin. Variables B
and Y are bound to each other and to atom lucy.
All the goals in the body of [RF1] have now succeeded, so the head
rich_father(gavin,lucy) succeeds, and in turn rich_father(A,B) succeds with A
and B bound to gavin and lucy, respectively.
Forcing the system to backtrack again will lead to the same sequence of
operations as above, right up to the attempt to find further matches for the goal
rich(X) in the body of [PF1]. This will fail, which will in turn cause [RF1] to fail.
This will make the Prolog system go back a further step to try to find another
match for the original goal rich_father(A,B) with clauses defining the
rich_father/2 predicate. Since there is only one such clause, no more matches will
be found and the user's goal will finally fail.
?- rich_father(A,B).
A = john ,
B = mary ;
A = john ,
B = mark ;
A = john ,
B = francis ;
A = gavin ,
B = lucy ;
?-
50
Logic Programming With Prolog
3.4 Satisfying Goals: A Summary
The method described in the previous sections is shown in diagrammatic form in
Figures 3.4 and 3.5. Note how the two flowcharts refer to each other.
Are there more
goals?
No
Sequence of goals
succeeds *
Yes
Succeeds *
Evaluate next goal
Fails
Are there any
previous goals?
No
Sequence of goals
fails
Yes
Fails
Re-evaluate
previous goal
Succeeds **
* Some variables may have become bound.
** Some variables may have become bound to other values (or unbound).
Figure 3.4 Evaluating a Sequence of Goals
Evaluating a Sequence of Goals: Summary
Evaluate the goals in turn, working from left to right. If they all succeed, the whole
sequence of goals succeeds. If one fails, go back through the previous goals in the
sequence one by one from right to left trying to resatisfy them. If they all fail, the
whole sequence fails. As soon as one succeeds, start working through the goals
from left to right again.
Satisfying Goals
Is predicate a BIP?
Yes
No
Are there more clauses
in the database? *
No
51
Goal is evaluated as
predefined by system
(may succeed or fail)
Goal fails
Yes
No
Can the goal be
unified with head of
next clause?
Yes **
Fails
Evaluate body of
clause (a sequence
of goals) ***
Succeeds **
Goal succeeds **
*
Evaluation: Start at top of database.
Re-evaluation: Start after clause matched when goal last satisfied.
** Some variables may have become bound.
*** If the clause is a fact there is no body, so the goal succeeds immediately.
Figure 3.5 Evaluating/Re-evaluating a Goal
Evaluating/Re-evaluating a Goal: Summary
Search through the clauses in the database, working from top to bottom* until one
is found, the head of which matches with the goal. If the matching clause is a fact,
the goal succeeds. If it is a rule, evaluate the sequence of goals in its body. If the
sequence succeeds, the goal succeeds. If not, continue searching through the
database for further matches. If the end of the database is reached, the goal fails.
* Start at the top (for evaluation) or after the clause matched when the goal was last
satisfied (for re-evaluation).
52
Logic Programming With Prolog
3.5 Removing Common Variables
When unifying a goal with a head of a clause, there is an important complication,
which all the examples in this chapter so far have been carefully designed to avoid:
what happens if the goal and the head have one or more variables in common?
Suppose that the goal is mypred(tuesday,likes(Z,Y),X) and the head is
mypred(X,Y,Z). The variables X, Y and Z in the former appear to be the same as
the variables X, Y and Z in the latter, but in fact there is no connection between
them. The clause of which mypred(X,Y,Z) is the head may be, for example,
mypred(X,Y,Z):-pred2(X,Q),pred3(Q,Y,Z).
The variables in this rule are just 'placeholders'. They can be replaced
consistently by any other variables without any change in the meaning of the
clause, as explained in Chapter 2, so it would not be sensible to consider X, Y and Z
in the clause to be the same variables as in the goal
mypred(tuesday,likes(Z,Y),X).
Before attempting to unify the goal and the head of the clause, it is first
necessary to rewrite the clause to ensure that it has no variables in common with
the goal. To be precise the clause must not have any variables in common with any
of the goals in the sequence of which the goal currently under consideration is part.
Prolog automatically replaces variables X, Y and Z in the clause systematically
by other variables that do not appear in the sequence of goals (or elsewhere in the
clause). For example they may be replaced by X1, Y1 and Z1.
mypred(X1,Y1,Z1):-pred2(X1,Q),pred3(Q,Y1,Z1).
After this rewriting it is only necessary to unify the head mypred(X1,Y1,Z1)
and the goal mypred(tuesday,likes(Z,Y),X), which have no variable in common.
The unification succeeds.
mypred(tuesday,likes(Z,Y),X)
mypred(X1,Y1,Z1):-pred2(X1,Q),pred3(Q,Y1,Z1).
Succeeds with variable X1 bound to atom tuesday, variable Y1 bound to
compound term likes(Z,Y) and variables Z1 and X bound to each other.
Variables Y and Z are unbound.
3.6 A Note on Declarative Programming
From this chapter it is clear that the order in which the clauses defining a predicate
occur in the database and the order of the goals in the body of a rule are of vital
importance when evaluating a user's query.
Satisfying Goals
53
It is part of the philosophy of logic programming that programs should be
written to minimize the effect of these two factors as far as possible. Programs that
do so are called fully or partly declarative.
An example of a fully declarative program is the following, based on Animals
Program 2 in Chapter 2.
dog(fido). dog(rover). dog(jane). dog(tom). dog(fred).
dog(henry).
cat(bill). cat(steve). cat(mary). cat(harry).
large(rover). large(william). large(martin).
large(tom). large(steve).
large(jim). large(mike).
large_animal(X):- dog(X),large(X).
large_animal(Z):- cat(Z),large(Z).
The query
?- large_animal(X).
will produce three possible values of X by backtracking
X = rover ;
X = tom ;
X = steve ;
no
Rearranging the clauses in the program in any order will produce the same
three answers but possibly in a different order (try it!).
Rearranging the order of the goals in the bodies of the two rules defining
large_animal/1, for example to
large_animal(X):- large(X),dog(X).
large_animal(Z):- large(Z),cat(Z).
will also give the same three answers.
It is often very difficult or impossible to define a predicate in such a way that the
order of the goals in the body of each rule is irrelevant, especially when built-in
predicates such as write/1 are involved. However, with a little effort it is
frequently possible to write the clauses defining a predicate in such a way that if
the order were changed the answers to any query (including those produced by
54
Logic Programming With Prolog
backtracking) would be the same. For example if we wish to test whether a number
is positive, negative or zero, we could define a predicate test/1 like this
test(X):-X>0,write(positive),nl.
test(0):-write(zero),nl.
test(X):-write(negative),nl.
This relies on the third clause only being reached when the value of X is negative.
A more declarative (and better) way of defining test/1 would be
test(X):-X>0,write(positive),nl.
test(0):-write(zero),nl.
test(X):-X<0,write(negative),nl.
where the test for X being negative in the third clause is made explicit.
Not only is it considered good Prolog programming style to make programs as
declarative as possible, it can greatly reduce the likelihood of making errors that
are hard to detect, particularly when backtracking is used. Chapter 7 gives some
examples of this.
Chapter Summary
This chapter demonstrates how Prolog uses unification to match goals with the
heads of clauses and how it uses the combination of unification and backtracking
to evaluate goals entered by the user and to find multiple solutions if required.
Satisfying Goals
55
Practical Exercise 3
The program below is a variant of the family relationships program used in Section
3.3. As before, [M1] etc. are labels added to make it easier to refer to the clauses.
[M1] mother(ann,henry).
[M2] mother(ann,mary).
[M3] mother(jane,mark).
[M4] mother(jane,francis).
[M5] mother(annette,jonathan).
[M6] mother(mary,bill).
[M7] mother(janice,louise).
[M8] mother(lucy,janet).
[M9] mother(louise,caroline).
[M10] mother(caroline,david).
[M11] mother(caroline,janet).
[F1] father(henry,jonathan).
[F2] father(john,mary).
[F3] father(francis,william).
[F4] father(francis,louise).
[F5] father(john,mark).
[F6] father(gavin,lucy).
[F7] father(john,francis).
[P1] parent(victoria,george).
[P2] parent(victoria,edward).
[P3] parent(X,Y):-mother(X,Y).
[P4] parent(X,Y):-father(X,Y).
[P5] parent(elizabeth,charles).
[P6] parent(elizabeth,andrew).
[A1] ancestor(X,Y):-parent(X,Y).
[A2] ancestor(X,Y):-parent(X,Z),ancestor(Z,Y).
The most important change is the addition of two clauses defining the ancestor/2
predicate. Clause [A1] simply states that X is an ancestor of Y if X is a parent of Y.
56
Logic Programming With Prolog
Clause [A2] is a recursive definition of more distant ancestor relationships which
can be read as 'X is the ancestor of Y if there is some person Z such that X is the
parent of Z and Z is the ancestor of Y'.
(1) Extend the program above by devising rules to define each of the following.
Load your extended program and test it.
(a) child_of(A,B)
(b) grandfather_of(A,B)
(c) grandmother_of(A,B)
(d) great_grandfather_of(A,B)
(2) Construct a sequence of diagrams similar to those in Section 3.3 to show the
sequence of events when the Prolog system attempts to satisfy the goal
?-ancestor(louise,Desc).
Find (using backtracking) the first two people the Prolog system will identify as
louise's descendants.
Predict the output that will be produced if the user repeatedly forces the system to
backtrack. Verify your prediction by loading the program and testing it.
4
Operators and Arithmetic
Chapter Aims
After reading this chapter you should be able to:
•
Convert unary and binary predicates to operators
•
Evaluate arithmetic expressions and compare their values
•
Test for equality of arithmetic expressions and terms
•
Use the 'not' and disjunction operators.
4.1 Operators
Up to now, the notation used for predicates in this book is the standard one of a
functor followed by a number of arguments in parentheses, e.g. likes(john,mary).
As an alternative, any user-defined predicate with two arguments (a binary
predicate) can be converted to an infix operator. This enables the functor
(predicate name) to be written between the two arguments with no parentheses, e.g.
john likes mary
Some Prolog users may find this easier to read. Others may prefer the standard
notation.
Any user-defined predicate with one argument (a unary predicate) can be
converted to a prefix operator. This enables the functor to be written before the
argument with no parentheses, e.g.
isa_dog fred
58
Logic Programming With Prolog
instead of
isa_dog(fred)
Alternatively, a unary predicate can be converted to a postfix operator. This
enables the functor to be written after the argument, e.g.
fred isa_dog
Operator notation can also be used with rules to aid readability. Some people
may find a rule such as
likes(john,X):-is_female(X),owns(X,Y),isa_cat(Y).
easier to understand if it is written as
john likes X:- X is_female, X owns Y, Y isa_cat.
The standard bracketed 'functor and arguments' notation, e.g. likes(john,X) can
still be used with operators if preferred. 'Mixed' notation is also permitted, e.g. if
likes/2, is_female/1, owns/2 and isa_cat/1 are all operators
likes(john,X):-is_female(X),X owns Y,isa_cat(Y).
is a valid form of the previous rule.
Any user-defined predicate with one or two arguments can be converted to an
operator by entering a goal using the op predicate at the system prompt. This
predicate takes three arguments, for example
?-op(150,xfy,likes).
The first argument is the 'operator precedence', which is an integer from 0
upwards. The range of numbers used depends on the particular implementation.
The lower the number, the higher the precedence. Operator precedence values are
used to determine the order in which operators will be applied when more than one
is used in a term. The most important practical use of this is for operators used for
arithmetic, as will be explained later. In most other cases it will suffice to use an
arbitrary value such as 150.
The second argument should normally be one of the following three atoms:
xfy meaning that the predicate is binary and is to be converted to an infix operator
fy meaning that the predicate is unary and is to be converted to an prefix operator
Operators and Artithmetic
59
xf meaning that the predicate is unary and is to be converted to a postfix operator
The third argument specifies the name of the predicate that is to be converted to
an operator.
A predicate can also be converted to an operator by placing a line such as
?-op(150,xfy,likes).
in a Prolog program file to be loaded using consult or reconsult. Note that the
prompt (the two characters ?-) must be included. When a goal is used in this way,
the entire line is known as a directive. In this case, the directive must be placed in
the file before the first clause that uses the operator likes.
Several built-in predicates have been pre-defined as operators. These include
relational operators for comparing numerical values, including < denoting 'less
than' and > denoting 'greater than'.
Thus the following are valid terms, which may be included in the body of a
rule:
X>4
Y<Z
A=B
Bracketed notation may also be used with built-in predicates that are defined as
operators, e.g. >(X,4) instead of X>4.
A list of the principal built-in operators is given in Appendix 2 for ease of
reference.
4.2 Arithmetic
Although the examples used in previous chapters of this book are non-numerical
(animals which are mammals etc.), Prolog also provides facilities for doing
arithmetic using a notation similar to that which will already be familiar to many
users from basic algebra.
This is achieved using the built-in predicate is/2, which is predefined as an infix
operator and thus is written between its two arguments.
The most common way of using is/2 is where the first argument is an unbound
variable. Evaluating the goal X is –6.5 will cause X to be bound to the number –6.5
and the goal to succeed.
The second argument can be either a number or an arithmetic expression e.g.
X is 6*Y+Z-3.2+P-Q/4 (* denotes multiplication).
Any variables appearing in an arithmetic expression must already be bound (as
a result of evaluating a previous goal) and their values must be numerical. Provided
60
Logic Programming With Prolog
they are, the goal will always succeed and the variable that forms the first
argument will be bound to the value of the arithmetic expression. If not, an error
message will result.
?- X is 10.5+4.7*2.
X = 19.9
?- Y is 10,Z is Y+1.
Y = 10 ,
Z = 11
Symbols such as + - * / in arithmetic expressions are a special type of infix
operator known as arithmetic operators. Unlike operators used elsewhere in Prolog
they are not predicates but functions, which return a numerical value.
As well as numbers, variables and operators, arithmetic expressions can include
arithmetic functions, written with their arguments in parentheses (i.e. not as
operators). Like arithmetic operators these return numerical values, e.g. to find the
square root of 36:
?- X is sqrt(36).
X=6
The arithmetic operator - can be used not only as a binary infix operator to
denote the difference of two numerical values, e.g. X-6, but also as a unary prefix
operator to denote the negative of a numerical value, e.g.
?- X is 10,Y is -X-2.
X = 10 ,
Y = -12
The table below shows some of the arithmetic operators and arithmetic
functions available in Prolog.
X+Y
X-Y
X*Y
X/Y
X//Y
X^Y
-X
abs(X)
sin(X)
cos(X)
max(X,Y)
sqrt(X)
the sum of X and Y
the difference of X and Y
the product of X and Y
the quotient of X and Y
the 'integer quotient' of X and Y (the result is truncated to the
nearest integer between it and zero)
X to the power of Y
the negative of X
the absolute value of X
the sine of X (for X measured in degrees)
the cosine of X (for X measured in degrees)
the larger of X and Y
the square root of X
Operators and Artithmetic
61
Example
?- X is 30,Y is 5,Z is X+Y+X*Y+sin(X).
X = 30 ,
Y=5,
Z = 185.5
Although the is predicate is normally used in the way described here, the first
argument can also be a number or a bound variable with a numerical value. In this
case, the numerical values of the two arguments are calculated. The goal succeeds
if these are equal. If not, it fails.
?- X is 7,X is 6+1.
X=7
?- 10 is 7+13-11+9.
no
?- 18 is 7+13-11+9.
yes
Unification
The previous description can be simplified by saying that the second argument of
the is/2 operator is evaluated and this value is then unified with the first argument.
This illustrates the flexibility of the concept of unification.
(a) If the first argument is an unbound variable, it is bound to the value of the
second argument (as a side effect) and the is goal succeeds.
(b) If the first argument is a number, or a bound variable with a numerical value, it
is compared with the value of the second argument. If they are the same, the is goal
succeeds, otherwise it fails.
If the first argument is an atom, a compound term, a list, or a variable bound to one
of these (none of which should happen), the outcome is implementation-dependent.
It is likely that an error will occur.
Note that a goal such as X is X+1 will always fail, whether or not X is bound.
?- X is 10,X is X+1.
no
To increase a value by one requires a different approach.
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Logic Programming With Prolog
/* Incorrect version */
increase(N):-N is N+1.
?- increase(4).
no
/*Correct version */
increase(N,M):-M is N+1.
?- increase(4,X).
X=5
Operator Precedence in Arithmetic Expressions
When there is more than one operator in an arithmetic expression, e.g. A+B*C-D,
Prolog needs a means of deciding the order in which the operators will be applied.
For the basic operators such as + - * and / it is highly desirable that this is the
customary 'mathematical' order, i.e. the expression A+B*C-D should be interpreted
as 'calculate the product of B and C, add it to A and then subtract D', not as 'add A
and B, then multiply by C and subtract D'. Prolog achieves this by giving each
operator a numerical precedence value. Operators with relatively high precedence
such as * and / are applied before those with lower precedence such as + and -.
Operators with the same precedence (e.g. + and -, * and /) are applied from left to
right. The effect is to give an expression such as A+B*C-D the meaning that a user
who is familiar with algebra would expect it to have, i.e. A+(B*C)-D.
If a different order of evaluation is required this can be achieved by the use of
brackets, e.g. X is (A+B)*(C-D). Bracketed expressions are always evaluated first.
Relational Operators
The infix operators =:= =\= > >= < =< are a special type known as
relational operators. They are used to compare the value of two arithmetic
expressions. The goal succeeds if the value of the first expression is equal to, not
equal to, greater than, greater than or equal to, less than or less than or equal to the
value of the second expression, respectively. Both arguments must be numbers,
bound variables or arithmetic expressions (in which any variables are bound to
numerical values).
?- 88+15-3=:=110-5*2.
yes
?- 100=\=99.
yes
Operators and Artithmetic
63
4.3 Equality Operators
There are three types of relational operator for testing equality and inequality
available in Prolog. The first type is used to compare the values of arithmetic
expressions. The other two types are used to compare terms.
Arithmetic Expression Equality =:=
E1=:=E2 succeeds if the arithmetic expressions E1 and E2 evaluate to the same
value.
?- 6+4=:=6*3-8.
yes
?- sqrt(36)+4=:=5*11-45.
yes
To check whether an integer is odd or even we can use the checkeven/1 predicate
defined below.
checkeven(N):-M is N//2,N=:=2*M.
?- checkeven(12).
yes
?- checkeven(23).
no
?- checkeven(-11).
no
?- checkeven(-30).
yes
The integer quotient operator // divides its first argument by its second and
truncates the result to the nearest integer between it and zero. So 12//2 is 6, 23//2 is
11, -11//2 is -5 and -30//2 is -15. Dividing an integer by 2 using // and multiplying
it by 2 again will give the original integer if it is even, but not otherwise.
Arithmetic Expression Inequality =\=
E1=\=E2 succeeds if the arithmetic expressions E1 and E2 do not evaluate to the
same value
?- 10=\=8+3.
yes
64
Logic Programming With Prolog
Terms Identical ==
Both arguments of the infix operator == must be terms. The goal Term1==Term2
succeeds if and only if Term1 is identical to Term2. Any variables used in the
terms may or may not already be bound, but no variables are bound as a result of
evaluating the goal.
?- likes(X,prolog)==likes(X,prolog).
X=_
?- likes(X,prolog)==likes(Y,prolog).
no
(X and Y are different variables)
?- X is 10,pred1(X)==pred1(10).
X = 10
?- X==0.
no
?- 6+4==3+7.
no
The value of an arithmetic expression is only evaluated when used with the is/2
operator. Here 6+4 is simply a term with functor + and arguments 6 and 4. This is
entirely different from the term 3+7.
Terms Not Identical \==
Term1\==Term2 tests whether Term1 is not identical to Term2. The goal succeeds
if Term1==Term2 fails. Otherwise it fails.
?- pred1(X)\==pred1(Y).
X=_,
Y=_
(The output signifies that both X and Y are unbound and are different variables.)
Terms Identical With Unification =
The term equality operator = is similar to == with one vital (and often very useful)
difference. The goal Term1=Term2 succeeds if terms Term1 and Term2 unify, i.e.
there is some way of binding variables to values which would make the terms
identical. If the goal succeeds, such binding actually takes place. Unification is
discussed in detail in Chapter 3.
Operators and Artithmetic
65
?- pred1(X)=pred1(10).
X = 10
(Variable X is bound to 10, which makes the two terms identical.)
?- likes(X,prolog)=likes(john,Y).
X = john ,
Y = prolog
(Binding X to the atom john and Y to the atom prolog makes the two terms
identical.)
?- X=0,X=:=0.
X=0
(X=0 causes X to be bound to 0. The goal X=:=0 succeeds, which confirms that
X now has the value zero.)
?- 6+4=3+7.
no
(For the reason explained under ==.)
?- 6+X=6+3.
X=3
(Binding X to 3 makes the two terms identical. They are both 6+3, not the
number 9.)
?- likes(X,prolog)=likes(Y,prolog).
X=Y=_
(Binding X and Y makes the terms identical.)
?- likes(X,prolog)=likes(Y,ada).
no
(No unification can make the atoms prolog and ada identical.)
Non-Unification Between Two Terms \=
The goal Term1\=Term2 succeeds if Term1=Term2 fails, i.e. the two terms cannot
be unified. Otherwise it fails.
?- 6+4\=3+7.
yes
66
Logic Programming With Prolog
?- likes(X,prolog)\=likes(john,Y).
no
(Because binding X to john and Y to prolog will make the terms identical.)
?- likes(X,prolog)\=likes(X,ada).
X=_
4.4 Logical Operators
This section gives a brief description of two operators that take arguments that are
call terms, i.e. terms that can be regarded as goals.
The not Operator
The prefix operator not/1 can be placed before any goal to give its negation. The
negated goal succeeds if the original goal fails and fails if the original goal
succeeds.
The following examples illustrate the use of not/1. It is assumed that the
database contains the single clause
dog(fido).
?- not dog(fido).
no
?- dog(fred).
no
?- not dog(fred).
yes
?- X=0,X is 0.
X=0
?- X=0,not X is 0.
no
The Disjunction Operator
The disjunction operator ;/2 (written as a semicolon character) is used to represent
'or'. It is an infix operator that takes two arguments, both of which are goals.
Goal1;Goal2 succeeds if either Goal1 or Goal2 succeeds.
?- 6<3;7 is 5+2.
yes
Operators and Artithmetic
67
?- 6*6=:=36;10=8+3.
yes
Chapter Summary
This chapter introduces operator notation for predicates and describes the operators
provided for evaluating and comparing the values of arithmetic expressions, for
testing for equality of either arithmetic expressions or terms and for testing for the
negation of a goal or the disjunction of two goals.
68
Logic Programming With Prolog
Practical Exercise 4
(1) This program is based on Animals Program 3, given in Chapter 2.
dog(fido). large(fido).
cat(mary). large(mary).
dog(rover). small(rover).
cat(jane). small(jane).
dog(tom). small(tom).
cat(harry).
dog(fred). large(fred).
cat(henry). large(henry).
cat(bill).
cat(steve). large(steve).
large(jim).
large(mike).
large_dog(X):- dog(X),large(X).
small_animal(A):- dog(A),small(A).
small_animal(B):- cat(B),small(B).
chases(X,Y):large_dog(X),small_animal(Y),
write(X),write(' chases '),write(Y),nl.
Convert the seven predicates used to operator form and test your revised program.
The output should be the same as the output from the program above. Include
directives to define the operators in your program.
(2) Define and test a predicate which takes two arguments, both numbers, and
calculates and outputs the following values: (a) their average, (b) the square root of
their product and (c) the larger of (a) and (b).
5
Input and Output
Chapter Aims
After reading this chapter you should be able to:
•
Use the built-in predicates that read from and write to either the user's
terminal (keyboard and screen) or a file, both term by term and characterby-character in your own programs
•
Use ASCII values to manipulate strings of characters.
Introduction
Prolog has facilities to enable input and output either of terms or of characters.
Using terms is simpler and will be described first. Initially, it will be assumed that
all output is to the user's screen and all input is from the user's keyboard. Input and
output using external files, e.g. on a hard disk or CD-ROM, will be described in
Section 5.7 onwards. Note that, like many other built-in predicates, those for input
and output described in this chapter are all unresatisfiable, i.e. they always fail
when backtracking.
5.1 Outputting Terms
The main built-in predicate provided for outputting terms is write/1, which has
already been used many times in this book.
The write/1 predicate takes a single argument, which must be a valid Prolog
term. Evaluating the predicate causes the term to be written to the current output
stream, which by default is the user's screen. (The meaning of current output
70
Logic Programming With Prolog
stream will be explained in Sections 5.7 and 5.8. At present it can simply be taken
to mean the user's screen.)
The built-in predicate nl/0 has also been used many times previously in this
book. It takes no arguments. Evaluating a nl goal causes a new line to be output to
the current output stream.
Examples
?- write(26),nl.
26
yes
?- write('a string of characters'),nl.
a string of characters
yes
?- write([a,b,c,d,[x,y,z]]),nl.
[a,b,c,d,[x,y,z]]
yes
?- write(mypred(a,b,c)),nl.
mypred(a,b,c)
yes
?- write('Example of use of nl'),nl,nl,write('end of example'),nl.
Example of use of nl
end of example
yes
Note that atoms that have to be quoted on input (e.g. 'Paul', 'hello world') are not
quoted when output using write. If it is important to output the quotes, the
writeq/1 predicate can be used. It is identical to write/1 except that atoms that
need quotes for input are output between quotes (other atoms are not).
?- writeq('a string of characters'),nl.
'a string of characters'
yes
?-writeq(dog),nl.
dog
yes
?- writeq('dog'),nl.
dog
yes
Input and Output
71
5.2 Inputting Terms
The built-in predicate read/1 is provided to input terms. It takes a single argument,
which must be a variable.
Evaluating it causes the next term to be read from the current input stream,
which by default is the user's keyboard. (The meaning of current input stream will
be explained in Sections 5.7 and 5.9. At present it can simply be taken to mean the
user's keyboard.)
In the input stream, the term must be followed by a dot ('.') and at least one
white space character, such as space or newline. The dot and white space
characters are read in but are not considered part of the term.
Note that for input from the keyboard (only) a prompt character such as a colon
will usually be displayed to indicate that user input is required. It may be necessary
to press the 'return' key before Prolog will accept the input. Both of these do not
apply to input from files.
When a read goal is evaluated, the input term is unified with the argument
variable. If the variable is unbound (which is usually the case) it is bound to the
input value.
?- read(X).
: jim.
X = jim
?- read(X).
: 26.
X = 26
?- read(X).
: mypred(a,b,c).
X = mypred(a,b,c)
?- read(Z).
: [a,b,mypred(p,q,r),[z,y,x]].
Z = [a,b,mypred(p,q,r),[z,y,x]]
?- read(Y).
: 'a string of characters'.
Y = 'a string of characters'
If the argument variable is already bound (which for most users is far more
likely to occur by mistake than by design), the goal succeeds if and only if the
input term is identical to the previously bound value.
72
Logic Programming With Prolog
?- X=fred,read(X).
: jim.
no
?- X=fred,read(X).
: fred.
X = fred
5.3 Input and Output Using Characters
Although input and output of terms is straightforward, the use of quotes and full
stops can be cumbersome and is not always suitable. For example, it would be
tedious to define a predicate (using read) which would read a series of characters
from the keyboard and count the number of vowels. A much better approach for
problems of this kind is to input a character at a time. To do this it is first necessary
to know about the ASCII value of a character.
All printing characters and many non-printing characters (such as space and
tab) have a corresponding ASCII (American Standard Code for Information
Interchange) value, which is an integer from 0 to 255.
The table below gives the numerical ASCII values corresponding to the main
printable characters and some others.
9
tab
10
end of
record
32
space
33
!
34
"
35
#
36
37
38
39
$
%
&
'
40
(
41
)
42
*
43
+
44
,
45
-
46
.
47
/
48-57 0 to 9
58
:
59
;
60
<
94
^
61
=
95
_
62
>
96
`
63
?
64
@
97a to z
122
6590
A to Z
91
[
125 }
92
\
126 ~
93
]
123 {
124 |
Characters whose ASCII value is less than or equal to 32 are known as white
space characters.
Input and Output
73
5.4 Outputting Characters
Characters are output using the built-in predicate put/1. The predicate takes a
single argument, which must be a number from 0 to 255 or an expression that
evaluates to an integer in that range.
Evaluating a put goal causes a single character to be output to the current
output stream. This is the character corresponding to the numerical value (ASCII
value) of its argument, for example
?- put(97),nl.
a
yes
?- put(122),nl.
z
yes
?- put(64),nl.
@
yes
5.5 Inputting Characters
Two built-in predicates are provided to input a single character: get0/1 and get/1.
The get0 predicate takes a single argument, which must be a variable. Evaluating a
get0 goal causes a character to be read from the current input stream. The variable
is then unified with the ASCII value of this character.
Note that for input from the keyboard (only) a prompt character such as a colon
will usually be displayed to indicate that user input is required. It may be necessary
to press the 'return' key before Prolog will accept the input. Both of these also
apply to the get predicate described below but do not apply to input from files.
Assuming the argument variable is unbound (which will usually be the case), it
is bound to the ASCII value of the input character.
?- get0(N).
:a
N = 97
?- get0(N).
:Z
N = 90
74
Logic Programming With Prolog
?- get0(M)
:)
M = 41
If the argument variable is already bound, the goal succeeds if and only if it has
a numerical value that is equal to the ASCII value of the input character.
?- get0(X).
:a
X = 97
?- M is 41,get0(M).
:)
M = 41
?- M=dog,get0(M).
:)
no
?- M=41.001,get0(M).
:)
no
The get predicate takes a single argument, which must be a variable. Evaluating
a get goal causes the next non-white-space character (i.e. character with an ASCII
value less than or equal to 32) to be read from the current input stream. The
variable is then unified with the ASCII value of this character in the same way as
for get0.
?- get(X).
:Z
X = 90
?- get(M).
:
Z
M = 90
5.6 Using Characters: Examples
The first example shows how to read in a series of characters from the keyboard
finishing with * and to output their corresponding ASCII values one per line (for
all characters excluding *).
The predicate readin is defined recursively. It causes a single character to be
input and variable X to be bound to its (numerical) ASCII value. The action taken
(the process(X) goal) depends on whether or not X has the value 42 signifying a *
Input and Output
75
character. If it has, the evaluation of the goal stops. If not, the value of X is output,
followed by a new line, followed by a further call to readin. This process goes on
indefinitely until a * character is read. (In the example below, the ASCII values of
characters P, r, o etc. are correctly shown to be 80, 114, 111 etc.)
readin:-get0(X),process(X).
process(42).
process(X):-X=\=42,write(X),nl,readin.
?- readin.
: Prolog Example*
80
114
111
108
111
103
32
69
120
97
109
112
108
101
yes
The next example is an extended version of the one above. This time the ASCII
values of the input characters are not output, but the number of characters
(excluding the *) is output. The count predicate is defined with two arguments
which can be read as 'the number of characters counted so far' and 'the total number
of characters before the *'.
go(Total):-count(0,Total).
count(Oldcount,Result):get0(X),process(X,Oldcount,Result).
process(42,Oldcount,Oldcount).
process(X,Oldcount,Result):X=\=42,New is Oldcount+1,count(New,Result).
?- go(T).
: The time has come the walrus said*
T = 33
76
Logic Programming With Prolog
?- go(T).
:*
T=0
The final example is a recursive program, based on the previous two, which
shows how to read in a series of characters ending with * and count the number of
vowels. Characters are read in one by one until a character with ASCII value 42
(signifying *) is encountered.
Here the two arguments of the count predicate can be interpreted as 'the
number of vowels so far' and 'the total number of vowels'. The three arguments of
the process predicate can be read as 'the ASCII value of an input character', 'the
number of vowels up to but not including that character' and 'the total number of
vowels', respectively.
The first two arguments of the processChar predicate can be interpreted in the
same way as for process, but the third argument is 'the number of vowels up to and
including the character (first argument)'.
Predicate vowel tests for one of the 10 possible vowels (five upper case and
five lower case), using their ASCII values.
go(Vowels):-count(0,Vowels).
count(Oldvowels,Totvowels):get0(X),process(X,Oldvowels,Totvowels).
process(42,Oldvowels,Oldvowels).
process(X,Oldvowels,Totalvowels):X=\=42,processChar(X,Oldvowels,New),
count(New,Totalvowels).
processChar(X,Oldvowels,New):-vowel(X),
New is Oldvowels+1.
processChar(X,Oldvowels,Oldvowels).
vowel(65).
/*
A
*/
vowel(69).
/*
E
*/
vowel(73).
/*
I */
vowel(79).
/*
O
*/
vowel(85).
/*
U
*/
vowel(97).
/*
a
*/
vowel(101).
/*
e
*/
vowel(105).
/*
i */
Input and Output
vowel(111).
/*
o
*/
vowel(117).
/*
u
*/
77
?- go(Vowels).
: In the beginning was the word*
Vowels = 8
?- go(Vowels).
: pqrst*
Vowels = 0
5.7 Input and Output Using Files
Prolog takes all input from the current input stream and writes all output to the
current output stream. By default both of these are the stream named user,
denoting the user's terminal, i.e. keyboard for input and screen for output.
The same facilities available for input and output from and to the user's
terminal either term by term or character by character are also available for input
and output from and to files (e.g. files on a hard disk or a CD-ROM).
The user may open and close input and output streams associated with any
number of named files but there can only be one current input stream and one
current output stream at any time. Note that no file can be open for both input and
output at the same time (except user) and that the user input and output streams
cannot be closed.
5.8 File Output: Changing the Current Output Stream
The current output stream can be changed using the tell/1 predicate. This takes a
single argument, which is an atom or variable representing a file name, e.g.
tell('outfile.txt').
Evaluating a tell goal causes the named file to become the current output
stream. If the file is not already open, a file with the specified name is first created
(any existing file with the same name is deleted).
Note that the file corresponding to the previous current output stream remains
open when a new current output stream is selected. Only the current output stream
can be closed (using the told predicate described below).
The default current output stream is user, i.e. the user's terminal. This value can
be restored either by using the told predicate or by tell(user).
The built-in predicate told/0 takes no arguments. Evaluating a told goal causes
the current output file to be closed and the current output stream to be reset to user,
i.e. the user's terminal.
The built-in predicate telling/1 takes one argument, which must be a variable
and will normally be unbound. Evaluating a telling goal causes the variable to be
bound to the name of the current output stream.
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Logic Programming With Prolog
Output to a File
Although the above definition of tell states that 'any existing file with the same
name is deleted', there is another possibility, which is important for some
applications, namely that the file is not deleted and any output is placed after the
end of the existing contents of the file. Both the 'overwrite' and the 'append' options
are likely to be available in any practical implementation of Prolog but may
involve using a different predicate (e.g. open) instead of or as well as tell. See the
documentation of your version of Prolog for details.
5.9 File Input: Changing the Current Input Stream
The current input stream can be changed using the see/1 predicate. This takes a
single argument, which is an atom or variable representing a file name, e.g.
see('myfile.txt').
Evaluating a see goal causes the named file to become the current input stream.
If the file is not already open it is first opened (for read access only). If it is not
possible to open a file with the given name, an error will be generated.
Note that the file corresponding to the previous current input stream remains
open when a new current input stream is selected. Only the current input stream
can be closed (using the seen predicate described below).
The default current input stream is user, i.e. the user's terminal. This value can
be restored either by using the seen predicate or by see(user).
The built-in predicate seen/0 takes no arguments. Evaluating a see goal causes
the current input file to be closed and the current input stream to be reset to user,
i.e. the user's terminal.
The built-in predicate seeing/1 takes one argument, which must be a variable
and will normally be unbound. Evaluating a seeing goal causes the variable to be
bound to the name of the current input stream.
5.9.1 Reading from Files: End of File
If the end of file is encountered when evaluating the goal read(X), variable X will
be bound to the atom end_of_file.
If the end of file is encountered while evaluating the goal get(X) or get0(X),
variable X will be bound to a 'special' numerical value. As ASCII values must be in
the range 0 to 255 inclusive, this will typically be -1, but may vary from one
implementation of Prolog to another.
Input and Output
79
5.9.2 Reading from Files: End of Record
Depending on the version of Prolog used, there may be an incompatibility for
character input between reading the end of a record (i.e. the character(s) that
signify the end of a line) from the user's terminal and from a file.
Typically the end of a line of input at the user's terminal will be indicated by
the character with ASCII value 13. The end of a record in a file will generally be
indicated by two ASCII values: 13 followed by 10.
The following program shows how to read in a series of characters from the
keyboard and print them out, one per line.
readline:-get0(X),process(X).
process(13).
process(X):-X=\=13,put(X),nl,readline.
Note the use of put rather than write and that the test for ASCII value 13
avoids the need for a character such as * to indicate 'end of input'.
?- readline.
: Prolog test
P
r
o
l
o
g
t
e
s
t
yes
5.10 Using Files: Examples
Example 1
Adapt the final program given in Section 5.6 to read the characters in a text file
myfile.txt until a * character is reached and output the number of vowels to the
user's terminal (i.e. the screen).
Only the first line of the previous program needs to be changed, to:
go(Vowels):-see('myfile.txt'),count(0,Vowels),seen.
80
Logic Programming With Prolog
Example 2
Define a predicate readterms to read the first four terms from a specified file and
output them to another specified file, one per line.
A suitable definition is given below.
readterms(Infile,Outfile):see(Infile),tell(Outfile),
read(T1),write(T1),nl,read(T2),write(T2),nl,
read(T3),write(T3),nl,read(T4),write(T4),nl,
seen,told.
Assuming the contents of file textfile.txt are the three lines:
'first term'. 'second term'.
'third term'.
'fourth term'. 'fifth term'.
using readterms gives the following brief output:
?- readterms('textfile.txt','outfile.txt').
yes
and creates a file with four lines of text
first term
second term
third term
fourth term
Although the definition of readterms above is correct as far as it goes, the final
two terms (seen and told) will cause the current input and output streams to be set
to user. This could cause problems if readterms were used as a subgoal in a larger
program where the current input and output streams were not necessarily both user
when it was called.
It is good programming practice to restore the original input and output streams
as the final steps when a goal such as readterms is evaluated. This can be achieved
for input by placing the goals seeing(S) and see(S) before and after the other terms
in the body of a rule. The former binds S to the name of the current input stream;
the latter resets the current input stream to S.
A similar effect can achieved for output by placing the goals telling(T) and
tell(T) before and after the other terms in the body of a rule. The former binds T to
the name of the current output stream; the latter resets the current output stream to
T.
Using these conventions, the revised definition of readterms is as follows:
Input and Output
81
readterms(Infile,Output):seeing(S),see(Infile),telling(T),tell(Outfile),
read(T1),write(T1),nl,read(T2),write(T2),nl,
read(T3),write(T3),nl,read(T4),write(T4),nl,
seen,see(S),told,tell(T).
Example 3
Define a predicate copychars to copy characters input (as a single line) at the
user's terminal to a specified file, until the character ! is entered (this character
should not be copied).
In the program below, copychars mainly saves and restores the values of the
current input and output streams. The rest of the task is left to copy_characters,
which is defined recursively in a similar way to readin in the first example in
Section 5.6.
copychars(Outfile):- telling(T),tell(Outfile),
copy_characters,told,tell(T).
copy_characters:-get0(N),process(N).
/*
33 is ASCII value of character ! */
process(33).
process(N):-N=\=33,put(N),copy_characters.
Using copychars as follows
?- copychars('myfile.txt').
: abxyz!
yes
will place the characters abxyz in file myfile.txt.
Chapter Summary
Describes the principal built-in predicates available for both term by term and
character by character input and output and for reading and writing files. Also
introduces the notion of the ASCII value of a character.
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Logic Programming With Prolog
Practical Exercise 5
(1) Define a predicate makelower/0 which reads in a line of characters from the
keyboard and outputs it again as a single line with any upper case letters converted
to lower case. (The ASCII values of the characters a, z, A and Z are 97, 122, 65 and
90, respectively.)
Thus the following would be a typical use of makelower:
?- makelower.
: This is an Example 123 inCLUDing numbers and symbols +-*/@[] XYz
this is an example 123 including numbers and symbols +-*/@[] xyz
yes
(2) Define a predicate copyterms which reads all the terms in a text file and
outputs them as terms to another text file one by one on separate lines.
The output file should be in a format suitable for use as the input file in a
subsequent call of copyterms. Thus for example if the input file contained
'first term'. 'second term'.
'third term'.
fourth. 'fifth term'.
sixth.
The output file would contain
'first term'.
'second term'.
'third term'.
fourth.
'fifth term'.
sixth.
(3) Create a text file testa.txt containing two lines, each of five characters followed
by a new line, e.g.
abcde
fghij
Define a predicate readfile that will read fifteen characters from this file one by
one and output the ASCII value of each character. Use this to establish whether the
Input and Output
83
representations of 'end of file' and 'end of record' for your version of Prolog are as
suggested in Sections 5.9.1 and 5.9.2, respectively.
(4) Using a text editor, create two text files in1.txt and in2.txt, each comprising a
number of terms terminated by end.
Define and test a predicate combine that takes the names of two input files as its
first two arguments and the name of an output file as its third argument. The output
file should contain the terms in the first input file followed by the terms in the
second, one per line and terminated by end.
(5) Define and test a predicate compare that reads in two text files term by term
and for each pair of corresponding terms outputs a message either saying that they
are the same or that they are different. Assume that both files contain the same
number of terms and that the final term in each is end.
6
Loops
Chapter Aims
After reading this chapter you should be able to:
•
Define a predicate which causes a sequence of goals to be evaluated
repeatedly, either a fixed number of times or until a specified condition is
satisfied
•
Define a predicate which searches a database to find all the clauses with a
specified property.
Introduction
Most conventional programming languages have a looping facility that enables a
set of instructions to be executed repeatedly either a fixed number of times or until
a given condition is met. Although, as was pointed out in the introduction to this
book, Prolog has no looping facilities, similar effects can be obtained that enable a
sequence of goals to be evaluated repeatedly. This can be done in a variety of
ways, using backtracking, recursion, built-in predicates, or a combination of these.
6.1 Looping a Fixed Number of Times
Many programming languages provide 'for loops' which enable a set of instructions
to be executed a fixed number of times. No such facility is available in Prolog
(directly), but a similar effect can be obtained using recursion, as shown in the
example programs below.
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Logic Programing With Prolog
Example 1
The following program outputs integers from a specified value down to 1.
loop(0).
loop(N):-N>0,write('The value is: '),write(N),nl,
M is N-1,loop(M).
The loop predicate is defined in terms of itself. The second clause can be
thought of as: 'to loop from N, first write the value of N, then subtract one to give
M, then loop from M'. This process clearly needs to be terminated and this is
achieved by the first clause: 'when the argument is zero, do nothing (and hence
stop)'. The first clause can be regarded as a terminating condition for the recursion.
?- loop(6).
The value is: 6
The value is: 5
The value is: 4
The value is: 3
The value is: 2
The value is: 1
yes
Note the use of the two goals M is N-1,loop(M) in the second clause for the
loop predicate. The obvious alternative of loop(N-1) will not work. Prolog only
evaluates expressions such as N-1 when evaluating goals with functor is or one of
the relational operators, as described in Chapter 4. If N-1 is used as an argument of
a predicate it is taken to mean the term with infix operator - (i.e. a minus sign) and
arguments N and 1. This is most unlikely to be what is intended!
Example 2
The next program outputs integers from First to Last inclusive.
/* output integers from First to Last inclusive */
output_values(Last,Last):- write(Last),nl,
write('end of example'),nl.
output_values(First,Last):-First=\=Last,write(First),
nl,N is First+1,output_values(N,Last).
Here output_values has two arguments, which can be read as 'output the
integers from First to Last inclusive'. The loop terminates when both arguments are
the same.
Loops
87
?- output_values(5,12).
5
6
7
8
9
10
11
12
end of example
yes
Example 3
Define a predicate to find the sum of the integers from 1 to N (say for N = 100).
It is natural to think of this procedurally, i.e. start with 1, then add 2, then add 3,
then add 4, … , then add 100. However the process is much easier to program if reexpressed declaratively in terms of itself.
The sum of the first 100 integers is the sum of the first 99 integers, plus 100.
The sum of the first 99 integers is the sum of the first 98 integers, plus 99.
The sum of the first 98 integers is the sum of the first 97 integers, plus 98.
…………………………………………………………………………….
The sum of the first 3 integers is the sum of the first 2 integers, plus 3.
The sum of the first 2 integers is the sum of the first 1 integers, plus 2.
The sum of the first 1 integers is one.
There are two distinct cases to consider: the general case: 'the sum of the first
N integers is the sum of the first N-1 integers, plus N' and the terminating case: 'the
sum of the first 1 integers is 1'. This leads directly to the recursive definition:
/* sum the integers from 1 to N (the first argument)
inclusive */
sumto(1,1).
sumto(N,S):-N>1,N1 is N-1,sumto(N1,S1),S is S1+N.
?- sumto(100,N).
N = 5050
?- sumto(1,1).
yes
88
Logic Programing With Prolog
Note that using the additional variable N1 for holding the value of N-1 is
essential. Writing sumto(N-1,S1) etc. instead would not work correctly. N-1 is a
term, not a numerical value.
Example 4
Define a predicate to output the squares of the first N integers, one per line.
This can most easily be programmed if first recast in a recursive form, as
follows.
To output the squares of the first N integers, output the squares of the first N-1 and
then output N2
To output the squares of the first N-1 integers, output the squares of the first N-2
and then output (N-1)2
To output the squares of the first N-2 integers, output the squares of the first N-3
and then output (N-2)2
……………………………………………………………………………………….
To output the squares of the first 3 integers, output the squares of the first 2 and
then output 32
To output the squares of the first 2 integers, output the squares of the first 1 and
then output 22
To output the squares of the first 1 integers, output the number 1
Here the general case is 'to output the squares of the first N integers, output the
squares of the first N-1 and then output N2' and the terminating case is 'to output
the squares of the first 1 integers, output the number 1'. This leads to the following
two-clause program.
/* output the first N squares, one per line */
writesquares(1):-write(1),nl.
writesquares(N):-N>1,N1 is N-1,writesquares(N1),
Nsq is N*N,write(Nsq),nl.
?- writesquares(6).
1
4
9
16
25
36
yes
Loops
89
Example 5
The following program reads the first 6 terms from a specified file and writes them
to the current output stream. It uses a 'counting down' method, in a similar way to
Example 1.
read_six(Infile):-seeing(S),see(Infile),
process_terms(6),seen,see(S).
process_terms(0).
process_terms(N):-N>0,read(X),write(X),nl,N1 is N-1,
process_terms(N1).
6.2 Looping Until a Condition Is Satisfied
Many languages have an 'until loop' which enables a set of instructions to be
executed repeatedly until a given condition is met. Again, no such facility is
available directly in Prolog, but a similar effect can be obtained in several ways.
6.2.1 Recursion
The first example below shows the use of recursion to read terms entered by the
user from the keyboard and output them to the screen, until end is encountered.
go:-loop(start). /* start is a dummy value used to get
the looping process started.*/
loop(end).
loop(X):-X\=end,write('Type end to end'),read(Word),
write('Input was '),write(Word),nl,loop(Word).
?- go.
Type end to end: university.
Input was university
Type end to end: of.
Input was of
Type end to end: portsmouth.
Input was portsmouth
Type end to end: end.
Input was end
yes
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Logic Programing With Prolog
Using the disjunction operator ;/2 which was defined in Section 4.4 the above
program can be rewritten as a single clause.
loop:-write('Type end to end'),read(Word),
write('Input was '),write(Word),nl,
(Word=end;loop).
The 'disjunctive goal' (Word=end;loop) succeeds if variable Word is bound to
the atom end. If not, the system attempts to satisfy the goal loop recursively.
?- loop.
Type end to end: university.
Input was university
Type end to end: of.
Input was of
Type end to end: portsmouth.
Input was portsmouth
Type end to end: end.
Input was end
yes
This recursive program repeatedly prompts the user to enter a term until either
yes or no is entered.
get_answer(Ans):-write('Enter answer to question'),
nl,get_answer2(Ans).
get_answer2(Ans):write('answer yes or no'),
read(A),
((valid(A),Ans=A,write('Answer is '),
write(A),nl);get_answer2(Ans)).
valid(yes). valid(no).
?- get_answer(Myanswer).
Enter answer to question
answer yes or no: maybe.
answer yes or no: possibly.
answer yes or no: yes.
Answer is yes
Myanswer = yes
Loops
91
6.2.2 Using the 'repeat' Predicate
Although it can often be used to great effect, recursion is not always the easiest
way to provide the types of looping required in Prolog programs. Another method
that is often used is based on the built-in predicate repeat.
The name of this predicate is really a misnomer. The goal repeat does not
repeat anything; it merely succeeds whenever it is called. The great value of repeat
is that it also succeeds (as many times as necessary) on backtracking. The effect of
this, as for any other goal succeeding, is to change the order of evaluating goals
from 'right to left' (i.e. backtracking) back to 'left-to-right'. This can be used to
create a looping effect, as shown in the examples below.
This program repeatedly prompts the user to enter a term until either yes or no
is entered. It is an alternative to the recursive program shown at the end of the
previous section. In this case it is debatable whether using repeat is an
improvement on using recursion, but the example is included for purposes of
illustration.
get_answer(Ans):write('Enter answer to question'),nl,
repeat,write('answer yes or no'),read(Ans),
valid(Ans),write('Answer is '),write(Ans),nl.
valid(yes). valid(no).
The first five goals in the body of get_answer will always succeed. Evaluating
the fifth goal: read(Ans) will prompt the user to enter a term. If the term input is
anything but yes or no, say unsure, the following goal valid(Ans) will fail. Prolog
will then backtrack over read(Ans) and write('answer yes or no'), both of which
are unresatisfiable, i.e. will always fail on backtracking.
Backtracking will then reach the predicate repeat and succeed, causing
evaluation to proceed forward (left-to-right) again, with write('answer yes or no')
and read(Ans) both succeeding, followed by a further evaluation of valid(Ans).
Depending on the value of Ans, i.e. the user's input, the valid(Ans) goal will
either fail, in which case Prolog will backtrack as far as repeat, as before, or it will
succeed in which case the final three goals write('Answer is'), write(Ans) and nl
will all succeed. The overall effect is that the two goals write('answer yes or no')
and read(Ans) are called repeatedly until the terminating condition valid(Ans) is
satisfied, effectively creating a loop between repeat and valid(Ans).
?- get_answer(X).
Enter answer to question
answer yes or no: unsure.
answer yes or no: possibly.
answer yes or no: no.
answer is no
X = no
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Logic Programing With Prolog
Goals to the left of repeat in the body of a clause will never be reached on
backtracking.
The next program reads a sequence of terms from a specified file and outputs
them to the current output stream until the term end is encountered.
readterms(Infile):seeing(S),see(Infile),
repeat,read(X),write(X),nl,X=end,
seen,see(user).
In a similar way to the previous program, this effectively defines a loop
between the goals repeat and X=end.
If file myfile.txt contains the lines
'first term'. 'second term'.
'third term'. 'fourth term'.
'fifth term'. 'sixth term'.
'seventh term'.
'eighth term'.
end.
calling readterms will produce the following output
?- readterms('myfile.txt').
first term
second term
third term
fourth term
fifth term
sixth term
seventh term
eighth term
end
yes
This program shows how to implement a menu structure which loops back
repeatedly to request more input. Entering go at the prompt causes Prolog to output
a menu from which the user can choose activities one at a time until option d is
chosen. Note that all inputs are terms and so must be followed by a full stop
character.
Loops
93
go:- write('This shows how a repeated menu works'),
menu.
menu:-nl,write('MENU'),nl,
write('a. Activity A'),nl,write('b. Activity B'),nl,
write('c. Activity C'),nl,write('d. End'),nl,
read(Choice),nl,choice(Choice).
choice(a):-write('Activity A chosen'),menu.
choice(b):-write('Activity B chosen'),menu.
choice(c):-write('Activity C chosen'),menu.
choice(d):-write('Goodbye!'),nl.
choice(_):-write('Please try again!'),menu.
?- go.
This shows how a repeated menu works
MENU
a. Activity A
b. Activity B
c. Activity C
d. End
: b.
Activity B chosen
MENU
a. Activity A
b. Activity B
c. Activity C
d. End
: xxx.
Please try again!
MENU
a. Activity A
b. Activity B
c. Activity C
d. End
: d.
Goodbye!
yes
94
Logic Programing With Prolog
6.3 Backtracking with Failure
As the name implies, the predicate fail always fails, whether on 'standard'
evaluation left-to-right or on backtracking. Advantage can be taken of this,
combined with Prolog's automatic backtracking, to search through the database to
find all the clauses with a specified property.
6.3.1 Searching the Prolog Database
Supposing the database contains clauses such as
dog(fido).
dog(fred).
dog(jonathan).
Each dog clause can be processed in turn using the alldogs predicate defined
below.
alldogs:-dog(X),write(X),write(' is a dog'),nl,fail.
alldogs.
Calling alldogs will cause dog(X) to be matched with the dog clauses in the
database. Initially X will be bound to fido and 'fido is a dog' will be output. The
final goal in the first clause of the alldogs predicate will then cause evaluation to
fail. Prolog will then backtrack over nl and the two write goals (all of which are
unresatisfiable) until it reaches dog(X). This goal will succeed for a second time
causing X to be bound to fred.
This process will continue until fido, fred and jonathan have all been output,
when evaluation will again fail. This time the call to dog(X) will also fail as there
are no further dog clauses in the database. This will cause the first clause for
alldogs to fail and Prolog to examine the second clause of alldogs. This will
succeed and evaluation will stop.
The effect is to loop through the database finding all possible values of X that
satisfy the goal dog(X).
?- alldogs.
fido is a dog
fred is a dog
jonathan is a dog
yes
Note the importance of the second clause of the alldogs predicate. It is there to
ensure that, after the database has been searched, the goal succeeds. With only the
first line, any call to alldogs will eventually fail.
Loops
95
alldogs:-dog(X),write(X),write(' is a dog'),nl,fail.
?- alldogs.
fido is a dog
fred is a dog
jonathan is a dog
no
The next program is designed to search a database containing clauses
representing the name, age, place of residence and occupation of a number of
people.
If the database contains these five clauses
person(john,smith,45,london,doctor).
person(martin,williams,33,birmingham,teacher).
person(henry,smith,26,manchester,plumber).
person(jane,wilson,62,london,teacher).
person(mary,smith,29,glasgow,surveyor).
The names of all the teachers can be found using the allteachers predicate.
allteachers:-person(Forename,Surname,_,_,teacher),
write(Forename),write(' '),write(Surname),nl,
fail.
allteachers.
The effect of using backtracking with failure in this case is to find all the
teachers in the database.
?- allteachers.
martin williams
jane wilson
yes
If the second clause of allteachers were omitted, both teachers would still be
found but the evaluation of allteachers would end with failure. This is of little or
no importance when a goal is entered at the system prompt, but if allteachers were
used as a goal in the body of a rule it would obviously be desirable to ensure that it
always succeeded.
It should be noted that it is not always necessary to use 'backtracking with
failure' to search the database. For example, the predicate somepeople/0 defined
96
Logic Programing With Prolog
below will find all the people in the database given previously, down to williams,
using only standard backtracking.
somepeople:-person(Forename,Surname,_,_,_),
write(Forename),write(' '),write(Surname),nl,
Surname=williams.
somepeople.
The goal Surname=williams succeeds if the variable Surname is bound to
williams. If not, it fails. The effect is to search the database down to and including
the person clause with second argument williams.
?- somepeople.
john smith
martin williams
yes
6.3.2 Finding Multiple Solutions
Backtracking with failure can also be used to find all the ways of satisfying a goal.
Suppose that a predicate findroute(Town1,Town2,Route) finds a route Route
between two towns Town1 and Town2. The details of this predicate are irrelevant
here. It may be assumed that Town1 and Town2 are atoms and that Route is a list.
Backtracking with failure can then be used to find all possible routes between
Town1 and Town2 and write out each one on a separate line, as follows:
find_all_routes(Town1,Town2):findroute(Town1,Town2,Route),
write('Possible route: '),write(Route),nl,fail.
find_all_routes(_,_).
Chapter Summary
This chapter describes how a set of goals can be evaluated repeatedly in Prolog,
either a fixed number of times or until a specified condition is met, and how
multiple solutions can be arrived at using the technique of 'backtracking with
failure'.
Loops
97
Practical Exercise 6
(1) Define a predicate to output the values of the squares of the integers from N1 to
N2 inclusive and test it with N1 = 6 and N2 = 12.
(2) Define and test a predicate to read in a series of characters input by the user and
output all of those before the first new line or ? character.
(3) Using the person clauses given in Section 6.3.1, find the professions of all
those over 40.
7
Preventing Backtracking
Chapter Aims
After reading this chapter you should be able to:
•
Use the cut predicate to prevent unwanted backtracking
•
Use 'cut with failure' to specify exceptions to general rules.
Introduction
Backtracking (as described in Chapter 3) is a fundamental part of the process by
which the Prolog system satisfies goals. However, it can sometimes be too
powerful and lead to inappropriate results. This chapter is about preventing the
Prolog system from backtracking using a built-in predicate called cut, which is
written as an exclamation mark ! character.
Before going on, it is worth issuing a warning. Many Prolog users see
preventing backtracking using 'cut' as being 'against the spirit of the language' and
some would even like to see it banned altogether! It is included in the language
(and in this book) because it can sometimes be very useful. When used badly, it
can also be a cause of programming errors that are very hard to find. The best
advice is probably to use it only sparingly and with care.
7.1 The Cut Predicate
We start by giving two examples of predicate definitions that appear correct but
give erroneous results when used with backtracking.
100 Logic Programming With Prolog
Example 1
The larger predicate takes the value of the larger of its first two arguments (which
are assumed to be numbers) and returns it as the value of the third.
larger(A,B,A):-A>B.
larger(A,B,B).
With the usual 'top to bottom' searching of clauses, the second clause can
reasonably be assumed to apply only when A is less than or equal to B. Testing the
definition with 8 and 6 as the first two arguments gives the correct answer.
?- larger(8,6,X).
X=8
However, if the user forces the system to backtrack at this stage, it will go on to
examine the second clause for larger and generate an incorrect second answer.
?- larger(8,6,X).
X=8;
X=6
?Example 2
The definition of predicate sumto/2 given below is a slightly modified version of
the one given in Chapter 6. It still appears to be correct, but has a serious flaw.
The goal sumto(N,S) causes the sum of the integers from 1 to N to be
calculated and returns the answer as the value of S.
sumto(1,1).
sumto(N,S):-N1 is N-1,sumto(N1,S1),
S is S1+N.
?- sumto(3,S).
S=6
However, forcing backtracking will now cause the system to crash with a
cryptic error message, such as 'stack overflow'. Whilst evaluating the goal
sumto(3,S) the Prolog system will try to find a solution for the goal sumto(1,S).
The first time it does this the first clause is used and the second argument is
correctly bound to 1. On backtracking the first clause is rejected and the system
attempts to satisfy the goal using the second clause. This causes it to subtract one
from one and then evaluate the goal sumto(0,S). Doing this will in turn require it
Preventing Backtracking 101
to evaluate sumto(-1,S1), then sumto(-2,S1) and so on, until eventually the system
runs out of memory.
Examples 1 and 2 could both be remedied by using additional goals in the
definition of the predicates, e.g. by changing the second clause of the definition of
larger to
larger(A,B,B):-A=<B.
and the second clause in the definition of sumto to
sumto(N,S):-N>1,N1 is N-1,sumto(N1,S1),S is
S1+N.
However, in other cases identifying such additional terms can be considerably
more difficult.
A more general way to avoid unwanted backtracking is to use a cut. The goal !
(pronounced 'cut') in the body of a rule always succeeds when first evaluated. On
backtracking it always fails and prevents any further evaluation of the current goal,
which therefore fails.
Example 1 (revised)
larger(A,B,A):-A>B,!.
larger(A,B,B).
?- larger(8,6,X).
X=8
?Example 2 (revised)
sumto(1,1):-!.
sumto(N,S):-N1 is N-1,sumto(N1,S1),
S is S1+N.
?- sumto(6,S).
S = 21
?Note that backtracking over a cut not only causes the evaluation of the current
clause of larger or sumto to be abandoned but also prevents the evaluation of any
other clauses for that predicate.
102 Logic Programming With Prolog
Example 3
The following incorrect program defines a predicate classify/2 that classifies a
number (its first argument) as either positive, negative or zero. The first clause
deals explicitly with the case where the first argument is zero. The second deals
with a negative value, leaving the third to deal with positive values.
/* classify a number as positive, negative or zero */
classify(0,zero).
classify(N,negative):-N<0.
classify(N,positive).
However, as before, the absence of a specific test for a positive argument
causes problems when the user forces the system to backtrack.
?- classify(0,N).
N = zero ;
N = positive
?- classify(-4,X).
X = negative ;
X = positive
?This can be rectified either by changing the third clause to
classify(N,positive):-N>0.
or by using cuts.
Example 3 (revised)
classify(0,zero):-!.
classify(N,negative):-N<0,!.
classify(N,positive).
?- classify(0,N).
N = zero
?- classify(-4,N).
N = negative
?-
Preventing Backtracking 103
So far all the incorrect programs could have been rectified by adding an
additional goal to one of the clauses rather than using cuts, and that would
probably have been the better approach. The following example shows a more
difficult case.
Example 4
A very common requirement is to prompt the user for an answer to a question until
a valid answer (e.g. yes or no) is entered. The following program does this using a
repeat loop, but unhelpfully will continue to prompt for valid answers on
backtracking.
get_answer(Ans):write('Enter answer to question'),nl,
repeat,write('answer yes or no'),read(Ans),
valid(Ans),write('Answer is '),write(Ans),nl.
valid(yes).
valid(no).
?- get_answer(X).
Enter answer to question
answer yes or no: maybe.
answer yes or no: yes.
Answer is yes
X = yes ;
answer yes or no: no.
Answer is no
X = no ;
answer yes or no: unsure.
answer yes or no: yes.
Answer is yes
X = yes
(and so on indefinitely).
Example 4 (revised)
Adding a final cut to the definition of get_answer will prevent the unwanted
backtracking.
get_answer(Ans):write('Enter answer to question'),nl,
104 Logic Programming With Prolog
repeat,write('answer yes or no'),read(Ans),
valid(Ans),write('Answer is '),write(Ans),nl,!.
valid(yes).
valid(no).
?- get_answer(X).
Enter answer to question
answer yes or no: maybe.
answer yes or no: unsure.
answer yes or no: yes.
Answer is yes
X = yes
?The above example, like all the other examples in this chapter so far, illustrates
the solution to a problem that could have been avoided if the user had not chosen to
force the system to backtrack. In practice, it is most unlikely that anyone would
want to do so. The prevention of unwanted backtracking is of much more practical
importance when one predicate 'calls' another (i.e. makes use of another as a goal
in the body of one of its clauses). This can lead to apparently inexplicable results.
Example 5
The go predicate in the following program uses a repeat loop to prompt the user
for input until a positive number is entered. However, the lack of cuts in the
definition of the classify predicate leads to incorrect answers.
classify(0,zero).
classify(N,negative):-N<0.
classify(N,positive).
go:-write(start),nl,
repeat,write('enter a positive value'),read(N),
classify(N,Type),Type=positive,
write('positive value is '),write(N),nl.
Preventing Backtracking 105
?- go.
start
enter a positive value: 28.
positive value is 28
yes
?- go.
start
enter a positive value: -6.
positive value is -6
yes
?- go.
start
enter a positive value: 0.
positive value is 0
yes
Changing the definition of classify to the one given in Example 3 (revised)
above gives the expected behaviour for go.
Example 5 (revised)
classify(0,zero):-!.
classify(N,negative):-N<0,!.
classify(N,positive).
go:-write(start),nl,
repeat,
write('enter a positive value'),read(N),
classify(N,Type),
Type=positive,
write('positive value is '),write(N),nl.
?- go.
start
enter a positive value: -6.
enter a positive value: -7.
enter a positive value: 0.
enter a positive value: 45.
positive value is 45
yes
106 Logic Programming With Prolog
7.2 Cut with Failure
Another use of 'cut' that can sometimes be helpful is to specify exceptions to
general rules.
Suppose that we have a database of the names of birds, such as
bird(sparrow).
bird(eagle).
bird(duck).
bird(crow).
bird(ostrich).
bird(puffin).
bird(swan).
bird(albatross).
bird(starling).
bird(owl).
bird(kingfisher).
bird(thrush).
A natural rule to add to this would be
can_fly(X):-bird(X).
corresponding to 'all birds can fly'.
Unfortunately this rule is over general. There are a few exceptions, notably that
ostriches cannot fly. How can we ensure that the goal can_fly(ostrich) will always
fail? The obvious approach is to change the definition of the can_fly predicate to
can_fly(ostrich):-fail.
can_fly(X):-bird(X).
However this does not give the desired result:
?- can_fly(duck).
yes
?- can_fly(ostrich).
yes
Preventing Backtracking 107
The can_fly(ostrich) goal is matched with the head of the first can_fly clause.
Attempting to satisfy the goal in the body of that clause (i.e. fail) obviously fails,
so the system next looks at the second can_fly clause. The goal matches with the
head, and the goal in the body of the clause, i.e. bird(X) is also satisfied, so the
can_fly(ostrich) goal succeeds. This is obviously not what was intended.
The desired effect can be achieved by replacing the can_fly clauses by
can_fly(ostrich):-!,fail.
can_fly(X):-bird(X).
?- can_fly(duck).
yes
?- can_fly(ostrich).
no
As before, the can_fly(ostrich) goal is matched with the head of the first
can_fly clause. Attempting to satisfy the goal in the body of that clause (i.e. fail)
fails, but the cut prevents the system from backtracking and so the
can_fly(ostrich) goal fails.
The combination of goals !,fail is known as cut with failure.
Chapter Summary
This chapter describes how the 'cut' predicate can be used to prevent undesirable
backtracking and how 'cut' can be used in conjunction with the 'fail' predicate to
specify exceptions to general rules.
108 Logic Programming With Prolog
Practical Exercise 7
(1) The predicate defined below is intended to correspond to the mathematical
function factorial. The factorial of a positive integer N is defined as the product of
all the integers from 1 to N inclusive, e.g. the factorial of 6 is 1 x 2 x 3 x 4 x 5 x 6
= 720.
factorial(1,1).
factorial(N,Nfact):-N1 is N-1,
factorial(N1,Nfact1),Nfact is N*Nfact1.
The definition of the factorial predicate is incorrect as given, and using it can
cause the system to crash.
Demonstrate that this definition is incorrect by entering a goal such as
factorial(6,N). Use backtracking to try to find more than one solution.
Correct the program and use it to find the factorials of 6 and 7.
(2) The following is part of a program that defines a predicate go which prompts
the user to input a series of numbers ending with 100 and outputs a message saying
whether each is odd or even.
go:-repeat,read_and_check(N,Type),
write(N),write(' is '),write(Type),nl,N=:=100.
Complete the program by defining the predicate read_and_check to obtain output
such as that given below. Your program should use at least one cut.
?- go.
Enter next number: 23.
23 is odd
Enter next number: -4.
-4 is even
Enter next number: 13.
13 is odd
Enter next number: 24.
24 is even
Enter next number: 100.
100 is even
yes
8
Changing the Prolog Database
Chapter Aims
After reading this chapter you should be able to:
•
Define a predicate which causes one or more clauses to be added to or
deleted from the Prolog database
•
Define predicates to create and manipulate a database of related facts
within the Prolog database.
8.1 Changing the Database: Adding and Deleting
Clauses
The normal way of placing clauses in the Prolog database is to consult or
reconsult one or more files. The difference between the two predicates was
explained in Chapter 2.
A consult directive causes all the clauses in the file to be loaded into the
database to add to those already there.
A reconsult directive behaves in a similar way to consult, with one crucial
difference. If any clause in the reconsulted file is for the same predicate as a clause
already in the database (i.e. their heads have the same functor and arity), all clauses
for that predicate in the database are first deleted.
Clauses placed into the database normally stay there until added to or deleted
by a subsequent consult or reconsult directive, or until the user exits from the
Prolog system when all clauses are automatically deleted. For most purposes this is
entirely sufficient. However Prolog also has built-in predicates for adding clauses
to and deleting clauses from the database which can be useful for more advanced
programming in the language. Like many other 'advanced' features, they need to be
110 Logic Programming With Prolog
used with care. As usual, these built-in predicates can be used either in the body of
a rule or in a directive entered at the system prompt.
As the user's program and the Prolog database are equivalent, using them in the
body of a clause is equivalent to modifying the user's program while it is being
used.
Specifying a Predicate as Dynamic
If any predicate is to be modified using assertz, retract etc., it should be specified
as dynamic by a directive in the user's program.
As an example, for predicate mypred with arity 3, the line
?-dynamic(mypred/3).
should be added at or near the start of the program and certainly before the first
clause that mentions the mypred predicate.
If the dynamic directive is left out, attempts to modify the database are likely to
produce error messages such as 'Predicate Protected' or even 'Predicate Not
Defined'.
8.2 Adding Clauses
Two main predicates are available for adding clauses to the database. Both take a
single argument, which must be a clause, i.e. a fact or a rule.
assertz(Clause)
The predicate assertz/1 causes Clause to be added to the database at the end of the
sequence of clauses that define the corresponding predicate.
The clause used for the first argument should be written without a terminating
full stop. Rules must be enclosed in an additional pair of parentheses, e.g.
?-assertz(dog(fido)).
?-assertz((go:-write('hello world'),nl)).
The clause may include one or more variables, e.g.
?-assertz(dog(X)).
?-assertz((go(X):-write('hello '),write(X),nl)).
asserta(Clause)
The predicate asserta/1 causes Clause to be added to the database at the start of the
sequence of clauses that define the corresponding predicate.
Changing the Prolog Database 111
The clause used for the first argument should be written without a terminating
full stop. Rules must be enclosed in an additional pair of parentheses, e.g.
?-asserta(dog(fido)).
?-asserta((go:-write('hello world'),nl)).
8.3 Deleting Clauses
Two main predicates are available for deleting clauses from the database.
retract(Clause)
The predicate retract/1 takes a single argument, which must be a clause, i.e. a fact
or a rule. It causes the first clause in the database that matches (i.e. unifies with)
Clause to be deleted.
If the following clauses are in the database
dog(jim).
dog(fido).
dog(X).
the query
?-retract(dog(fido)).
will delete the second clause and the further query
?-retract(dog(X)).
will delete the dog(jim) clause, which is the first one of the remaining two clauses
to unify with the query.
This will leave the dog(X) clause in the database. Although unusual, this is a
valid Prolog fact which signifies 'everything is a dog'.
retractall(Head)
The predicate retractall/1 takes a single argument which must be the head of a
clause. It causes every clause in the database whose head matches Head to be
deleted. The retractall goal always succeeds even if no clauses are deleted.
Some examples are:
?-retractall(mypred(_,_,_)).
which deletes all the clauses for the mypred/3 predicate, and
112 Logic Programming With Prolog
?-retractall(parent(john,Y)).
which deletes all clauses for the parent/2 predicate which have the atom john as
their first argument.
Note that the directive
?-retractall(mypred).
only removes the clauses for predicate mypred/0. To delete all the clauses for
predicate mypred/3 it is necessary to use
?-retractall(mypred(_,_,_)).
with three anonymous variables.
8.4 Changing the Database: Example
The following example, which comprises a series of goals entered at the system
prompt, illustrates the use of the assertz, asserta, retract and retractall predicates
for changing the database.
?- listing(mypred).
yes
Initially there is no predicate mypred in
the database
?- assertz(mypred(first)).
yes
?- assertz(mypred(second)).
yes
?- assertz(mypred(third)).
yes
?- assertz(mypred(fourth)).
yes
Four mypred clauses are added to the
database
?- listing(mypred).
/* mypred/1 */
mypred( first ).
mypred( second ).
mypred( third ).
mypred( fourth ).
yes
?- asserta(mypred(new1)).
yes
The four mypred clauses are now in the
database
A new mypred clause is added to the
database, using the asserta predicate
Changing the Prolog Database 113
?- listing(mypred).
/* mypred/1 */
mypred( new1 ).
mypred( first ).
mypred( second ).
mypred( third ).
mypred( fourth ).
yes
?- assertz(mypred(new2)).
yes
This shows that asserta places the new
clause above the other mypred clauses in
the database
A further mypred clause is added to the
database using the assertz predicate
?- listing(mypred).
/* mypred/1 */
mypred( new1 ).
mypred( first ).
mypred( second ).
mypred( third ).
mypred( fourth ).
mypred( new2 ).
yes
?- mypred(X).
X = new1 ;
X = first ;
X = second ;
X = third ;
X = fourth ;
X = new2
?- retract(mypred(first)).
yes
This shows that assertz places the new
clause below the other mypred clauses in
the database
This shows that retrieving mypred
clauses from the database will give the
first one listed by the listing predicate
Subsequent backtracking will obtain the
remaining clauses in top to bottom order
One of the mypred clauses is removed
from the database using the retract
predicate
?- listing(mypred).
/* mypred/1 */
mypred( new1 ).
mypred( second ).
mypred( third ).
mypred( fourth ).
mypred( new2 ).
yes
This shows that it has successfully been
removed
114 Logic Programming With Prolog
?- retractall(mypred).
yes
This is an attempt to remove all the
mypred clauses from the database, but is
incorrect
?- listing(mypred).
/* mypred/1 */
mypred( new1 ).
mypred( second ).
mypred( third ).
mypred( fourth ).
mypred( new2 ).
This shows that the database is unaltered
?- retractall(mypred(_)).
yes
This is the correct way to remove all
mypred clauses with one argument from
the database
?- listing(mypred).
yes
This shows that all the mypred clauses
have now been removed
8.5 Maintaining a Database of Facts
The predicates assertz, retract etc. can be used to create and maintain a database
of related facts within the full Prolog database of facts and rules.
Creating a Database
Assume file people.txt contains the six lines
john. smith. 45. london. doctor.
martin. williams. 33. birmingham. teacher.
henry. smith. 26. manchester. plumber.
jane. wilson. 62. london. teacher.
mary. smith. 29. glasgow. surveyor.
end.
Predicate setup defined below will cause the file to be read and five clauses
such as
person(john,smith,45,london,doctor).
to be added to the Prolog database. The effect is the same as if the five clauses had
been included in the original program.
Changing the Prolog Database 115
setup:-seeing(S),see('people.txt'),
read_data,
write('Data read'),nl,
seen,see(S).
read_data:read(A),process(A).
process(end).
process(A):read(B),read(C),read(D),read(E),
assertz(person(A,B,C,D,E)),read_data.
?- setup.
Data read
yes
The listing predicate can be used to show all the clauses defining the person
predicate.
?- listing(person).
/* person/5 */
person( john, smith, 45, london, doctor ).
person( martin, williams, 33, birmingham, teacher ).
person( henry, smith, 26, manchester, plumber ).
person( jane, wilson, 62, london, teacher ).
person( mary, smith, 29, glasgow, surveyor ).
yes
Removing a Clause
Predicate remove will delete a single clause from the database.
remove(Forename,Surname):retract(person(Forename,Surname,_,_,_)).
?- remove(henry,smith).
yes
116 Logic Programming With Prolog
?- listing(person).
/* person/5 */
person( john, smith, 45, london, doctor ).
person( martin, williams, 33, birmingham, teacher ).
person( jane, wilson, 62, london, teacher ).
person( mary, smith, 29, glasgow, surveyor ).
yes
Changing a Clause
Predicate change will change a clause by retracting the old version and asserting a
new one.
change(Forename,Surname,New_Profession):person(Forename,Surname,Age,City,Profession),
retract(person(Forename,Surname,Age,City,Profession)),
assertz(person(Forename,Surname,Age,City,New_Profession)).
?- change(jane,wilson,architect).
yes
?- listing(person).
/* person/5 */
person( john, smith, 45, london, doctor ).
person( martin, williams, 33, birmingham, teacher ).
person( mary, smith, 29, glasgow, surveyor ).
person( jane, wilson, 62, london, architect ).
yes
Outputting the Database to a File
Predicate output_data will write out the person clauses to a new file in the same
format as the original file.
output_data:telling(T),tell('people2.txt'),
write_data,told,tell(T),
write('Data written'),nl.
Changing the Prolog Database 117
write_data:-person(A,B,C,D,E),
write(A),write('. '),
write(B),write('. '),
write(C),write('. '),
write(D),write('. '),
write(E),write('.'),nl,
fail.
write_data:-write('end.'),nl.
?- output_data.
Data written
yes
creates file people2.txt
john. smith. 45. london. doctor.
martin. williams. 33. birmingham. teacher.
mary. smith. 29. glasgow. surveyor.
jane. wilson. 62. london. architect.
end.
Chapter Summary
This chapter describes the built-in predicates for adding clauses to or deleting
clauses from the Prolog database and shows how to use them to create and
maintain a database of related facts within the overall Prolog database.
118 Logic Programming With Prolog
Practical Exercise 8
(1) Define and test a predicate add_data which reads a series of names of animals
(e.g. cat, dog, mouse) entered by the user, terminated by end. It should add a
corresponding series of facts, e.g. animal(dog) to the Prolog database, ignoring
any names that have already been entered.
(2) Define and test a predicate display_animals which lists the names of all the
animals in the database, one per line.
(3) Define and test a predicate remove2 which removes any clauses corresponding
to either a dog or a cat from the database, if they are present, and otherwise has no
effect.
9
List Processing
Chapter Aims
After reading this chapter you should be able to:
•
Represent data in the form of lists
•
Use built-in predicates to manipulate lists
•
Define predicates to work through a list element by element from left to
right using recursion.
9.1 Representing Data as Lists
Prolog's compound terms give a flexible way of representing data, especially as the
arguments may be other terms of any complexity. For example, the following is a
valid term: mypred(a,-6.3,pred2(p,pred3(3,q,aaa),r)).
However, in common with most other programming languages, compound
terms suffer from the limitation that each predicate must have a fixed number of
arguments. It is possible to use mypred sometimes with three arguments and
sometimes with four, say, but Prolog regards these as two entirely different
predicates.
To overcome this limitation, Prolog provides a very flexible type of data object
called a list. A list is written as a sequence of values, known as list elements,
separated by commas and enclosed in square brackets, e.g. [dog,cat,fish,man].
A list element does not have to be an atom. It can be any Prolog term, including
a bound or unbound variable or another list, so [X,Y,mypred(a,b,c),[p,q,r],z] is a
valid list. A list element that is itself a list is known as a sublist.
Lists can have any number of elements, including zero. The list with no
elements is known as the empty list and is written as [].
120 Logic Programming With Prolog
For non-empty lists, the first element is known as the head. The list remaining
after the first element is removed is called the tail. For example, the head of the list
[dog,cat,fish,man] is the atom dog and the tail is the list [cat,fish,man].
The head of list [x,y,mypred(a,b,c),[p,q,r],z] is the atom x. The tail is the list
[y,mypred(a,b,c),[p,q,r],z].
Some further examples of lists are:
[john,mary,10,robert,20,jane,X,bill]
[[john,28],[mary,56,teacher],robert,parent(victoria,albert),[a,b,[c,d,e],f],28]
[[portsmouth,london,glasgow],[portsmouth,london,edinburgh],[glasgow]]
9.2 Notation for Lists
Up to now lists have been written as a sequence of list elements written in order,
separated by commas and enclosed in square brackets. We will call this 'standard
bracketed notation'.
Lists are generally, although not always, written in this notation in queries
entered by the user at the system prompt, for example
?- X=alpha,Y=27,Z=[alpha,beta],write('List is: '),write([X,Y,Z]),nl.
List is: [alpha,27,[alpha,beta]]
However (with the exception of the empty list) lists are seldom written in this
way in a Prolog program. The explanation for this is that lists are most useful (and
generally only used) when the programmer does not know in advance how many
elements they will contain. If we know that a list will always contain three
elements, say a person's forename, surname and nationality, it would generally be
better to use a compound term with three arguments, such as person(john, smith,
british).
Lists are most valuable when the number of elements needed cannot be known
in advance and would probably vary from one use of the program to another. For
example, we might want to define a predicate that reads in information about an
organisation's purchases in a given financial year and writes out a list of all the
items of computer hardware or software purchased in the months from March to
June inclusive that cost more than a certain amount. In this case we would certainly
not wish to build in any assumption that the list will always have fifteen or any
other fixed number of elements. It might potentially be any number, from zero
upwards.
We need an alternative way of representing a list in a Prolog clause that does
not make any commitment to the number of elements it will have when the clause
is used. This is provided by the 'cons' (standing for list constructor) notation.
In this notation a list is written in a more complicated form than before, with
two parts joined together by the vertical bar character | which is known as the cons
List Processing 121
character or simply as cons. Thus a list is represented by the notation [ elements |
list ].
The elements part is a sequence of one or more list elements, which may be any
Prolog terms. The list part (obviously) represents a list.
The list [ elements | list ] is an augmented version of the list list with the
sequence of elements indicated by elements inserted before any elements that are
already there.
For example, [one|[two,three]] represents [one,two,three].
The following shows some equivalent ways of writing the same list of four
elements. We will come on to lists of variable length after that.
[alpha,beta,gamma,delta]
[alpha|[beta,gamma,delta]]
[alpha,beta|[gamma,delta]]
[alpha,beta,gamma|[delta]]
[alpha,beta,gamma,delta|[]]
[alpha,beta|[gamma|[delta|[]]]]
The cons notation for a list can be used anywhere the 'standard bracketed form'
would be valid, e.g.
?- write([alpha|[beta,gamma,delta]]),nl.
[alpha,beta,gamma,delta]
yes
?- write([alpha,beta,gamma|[delta]]),nl.
[alpha,beta,gamma,delta]
yes
?- write([alpha,beta,gamma,delta|[]]),nl.
[alpha,beta,gamma,delta]
yes
?- write([alpha,beta|[gamma,delta]]),nl.
[alpha,beta,gamma,delta]
yes
?- write([alpha,beta|[gamma|[delta|[]]]]),nl.
[alpha,beta,gamma,delta]
yes
122 Logic Programming With Prolog
In the common case where the elements part of [ elements | list ] consists of just
one term, the 'cons' notation can be used to construct a list from its head and tail,
which are the parts to the left and right of the vertical bar respectively. Thus
[a|[b,c,d]] denotes the list [a,b,c,d].
As illustrated so far, there would be no benefit gained by using the 'cons'
notation rather than the standard bracketed notation. The former is of most value
when the list part is a variable and/or the elements part contains one or more
variables.
For example, if variable L is bound to a list, say [red,blue,green,yellow], we
can represent a new list with the atom brown inserted before the elements already
there by [brown|L].
?- L=[red,blue,green,yellow],write([brown|L]),nl.
[brown,red,blue,green,yellow]
If variable A is bound to the list [brown,pink] and variable L is bound to the
list [red,blue,green,yellow], the list [A,A,black|L] represents
[[brown,pink],[brown,pink],black,red,blue,green,yellow].
We are now in a position to write a clause or a query that makes use of a list
without knowing in advance how many elements it will have. This example shows
a new list L1 created from a list L input by the user.
?- write('Type a list'),read(L),L1=[start|L],write('New list is '),write(L1),nl.
Type a list: [[london,paris],[x,y,z],27].
New list is [start,[london,paris],[x,y,z],27]
The 'cons' notation for lists is so much more flexible than the standard
bracketed notation that some would say that it is the 'correct' notation for lists, and
that a list written in the standard bracketed notation, such as [dog,cat,fish,man] is
just a more human-readable version of [dog|[cat|[fish|[man|[]]]]].
9.3 Decomposing a List
A common requirement is to perform the same (or a similar) operation on every
element of a list. By far the best way of processing a list is to work through its
elements one by one from left to right. This can be achieved by breaking the list
into its head and tail and processing each separately in a recursive fashion.
Paradoxically, breaking a list into its head and tail is often done using the list
constructor.
The predicate writeall defined below writes out the elements of a list, one per
line.
List Processing 123
The second clause of writeall separates a list into its head A and tail L, writes
out A and then a newline, then calls itself again recursively. The first clause of
writeall ensures that evaluation terminates when no further elements of the list
remain to be output.
/* write out the elements of a list, one per line */
writeall([]).
writeall([A|L]):- write(A),nl,writeall(L).
?- writeall([alpha,'this is a string',20,[a,b,c]]).
alpha
this is a string
20
[a,b,c]
yes
This definition of writeall is typical of many user-defined predicates for list
processing. Note that although writeall takes a list as its argument, its definition
does not include a statement beginning
writeall(L):Instead, the main part of the definition begins
writeall([A|L]):This makes the definition of the predicate considerably easier.
When a goal such as writeall([a,b,c]) is evaluated, it is matched against
(unified with) the head of the second clause of writeall. As this is written as
writeall([A|L]) the matching process causes A to be bound to atom a and L to be
bound to list [b,c]. This makes it easy for the body of the rule to process the head
and tail separately.
The recursive call to writeall with the tail of the original list, i.e. L, as its
argument is a standard programming technique used in list processing. As is
frequently the case, the empty list is treated separately, in this case by the first
clause of writeall.
The predicate write_english defined below takes as its argument a list such as
[[london,england],[paris,france],[berlin,germany],[portsmouth,england],
[bristol,england],[edinburgh,scotland]]
Each element is a sublist containing the name of a city and the name of the
country in which it is located. Calling write_english causes the names of all the
cities that are located in England to be output.
124 Logic Programming With Prolog
The second clause of write_english deals with those sublists that have the atom
england as their second element. In this case the first element is output, followed
by a new line and a recursive call to write_english, with the tail of the original list
as the argument. Sublists that do not have england as their second element are dealt
with by the final clause of write_english, which does nothing with the sublist but
makes a recursive call to write_english, with the tail of the original list as the
argument.
write_english([]).
write_english([[City,england]|L]):write(City),nl,
write_english(L).
write_english([A|L]):-write_english(L).
go:- write_english([[london,england],[paris,france],
[berlin,germany],[portsmouth,england],
[bristol,england],
[edinburgh,scotland]]).
?- go.
london
portsmouth
bristol
yes
The predicate replace defined below takes as its first argument a list of at least
one element. If the second argument is an unbound variable, it is bound to the same
list with the first element replaced by the atom first. Using the 'cons' notation, the
definition takes only one clause.
replace([A|L],[first|L]).
?- replace([1,2,3,4,5],L).
L = [first,2,3,4,5]
?- replace([[a,b,c],[d,e,f],[g,h,i]],L).
L = [first,[d,e,f],[g,h,i]]
List Processing 125
9.4 Built-in Predicate: member
The ability to represent data in the form of lists is such a valuable feature of Prolog
that several built-in predicates have been provided for it. The most commonly used
of these are described in this and the following sections.
The member built-in predicate takes two arguments. If the first argument is
any term except a variable and the second argument is a list, member succeeds if
the first argument is a member of the list denoted by the second argument (i.e. one
of its list elements).
?- member(a,[a,b,c]).
yes
?- member(mypred(a,b,c),[q,r,s,mypred(a,b,c),w]).
yes
?- member(x,[]).
no
?- member([1,2,3],[a,b,[1,2,3],c]).
yes
If the first argument is an unbound variable, it is bound to an element of the list
working from left to right (thus if it is called only once it will be bound to the first
element). This can be used in conjunction with backtracking to find all the
elements of a list in turn from left to right, as follows.
?- member(X,[a,b,c]).
X=a;
X=b;
X=c;
no
Predicate get_answer2 defined below reads a term entered by the user. It loops
using repeat, until one of a list of permitted answers (yes, no or maybe) is entered
and the member goal is satisfied.
get_answer2(Ans):-repeat,
write('answer yes, no or maybe'),read(Ans),
member(Ans,[yes,no,maybe]),
write('Answer is '),write(Ans),nl,!.
?- get_answer2(X).
answer yes, no or maybe: possibly.
answer yes, no or maybe: unsure.
answer yes, no or maybe: maybe.
answer is maybe
X = maybe
126 Logic Programming With Prolog
9.5 Built-in Predicate: length
The length built-in predicate takes two arguments. The first is a list. If the second
is an unbound variable it is bound to the length of the list, i.e. the number of
elements it contains.
?- length([a,b,c,d],X).
X=4
?- length([[a,b,c],[d,e,f],[g,h,i]],L).
L=3
?- length([],L).
L=0
If the second argument is a number, or a variable bound to a number, its value
is compared with the length of the list.
?- length([a,b,c],3).
yes
?- length([a,b,c],4).
no
?- N is 3,length([a,b,c],N).
N=3
9.6 Built-in Predicate: reverse
The reverse built-in predicate takes two arguments. If the first is a list and the
second is an unbound variable (or vice versa), the variable will be bound to the
value of the list with the elements written in reverse order, e.g.
?- reverse([1,2,3,4],L).
L = [4,3,2,1]
?- reverse(L,[1,2,3,4]).
L = [4,3,2,1]
?- reverse([[dog,cat],[1,2],[bird,mouse],[3,4,5,6]],L).
L = [[3,4,5,6],[bird,mouse],[1,2],[dog,cat]]
Note that the order of the elements of the sublists [dog,cat] etc. is not reversed.
If both arguments are lists, reverse succeeds if one is the reverse of the other.
?- reverse([1,2,3,4],[4,3,2,1]).
yes
List Processing 127
?- reverse([1,2,3,4],[3,2,1]).
no
The predicate front/2 defined below takes a list as its first argument. If the
second argument is an unbound variable it is bound to a list which is the same as
the first list with the last element removed. For example if the first list is [a,b,c],
the second will be [a,b]. In the body of the rule the first list L1 is reversed to give
L3. Its head is then removed to give L4 and L4 is then reversed back again to give
L2.
front(L1,L2):reverse(L1,L3),remove_head(L3,L4),reverse(L4,L2).
remove_head([A|L],L).
?- front([a,b,c],L).
L = [a,b]
?- front([[a,b,c],[d,e,f],[g,h,i]],L).
L = [[a,b,c],[d,e,f]]
The front predicate can also be used with two lists as arguments. In this case it
tests whether the second list is the same as the first list with the last element
removed.
?- front([a,b,c],[a,b]).
yes
?- front([[a,b,c],[d,e,f],[g,h,i]],[[a,b,c],[d,e,f]]).
yes
?- front([a,b,c,d],[a,b,d]).
no
9.7 Built-in Predicate: append
The term concatenating two lists means creating a new list, the elements of which
are those of the first list followed by those of the second list. Thus concatenating
[a,b,c] with [p,q,r,s] gives the list [a,b,c,p,q,r,s]. Concatenating [] with [x,y] gives
[x,y].
The append built-in predicate takes three arguments. If the first two arguments
are lists and the third argument is an unbound variable, the third argument is bound
to a list comprising the first two lists concatenated, e.g.
128 Logic Programming With Prolog
?- append([1,2,3,4],[5,6,7,8,9],L).
L = [1,2,3,4,5,6,7,8,9]
?- append([],[1,2,3],L).
L = [1,2,3]
?- append([[a,b,c],d,e,f],[g,h,[i,j,k]],L).
L = [[a,b,c],d,e,f,g,h,[i,j,k]]
The append predicate can also be used in other ways. When the first two
arguments are variables and the third is a list it can be used with backtracking to
find all possible pairs of lists which when concatenated give the third argument, as
follows.
?- append(L1,L2,[1,2,3,4,5]).
L1 = [] ,
L2 = [1,2,3,4,5] ;
L1 = [1] ,
L2 = [2,3,4,5] ;
L1 = [1,2] ,
L2 = [3,4,5] ;
L1 = [1,2,3] ,
L2 = [4,5] ;
L1 = [1,2,3,4] ,
L2 = [5] ;
L1 = [1,2,3,4,5] ,
L2 = [] ;
no
This example shows a list broken up in a more complex way.
?- append(X,[Y|Z],[1,2,3,4,5,6]).
X = [] ,
Y=1,
Z = [2,3,4,5,6] ;
X = [1] ,
Y=2,
Z = [3,4,5,6] ;
List Processing 129
X = [1,2] ,
Y=3,
Z = [4,5,6] ;
X = [1,2,3] ,
Y=4,
Z = [5,6] ;
X = [1,2,3,4] ,
Y=5,
Z = [6] ;
X = [1,2,3,4,5] ,
Y=6,
Z = [] ;
no
9.8 List Processing: Examples
This section shows some examples of list processing, all of which illustrate the use
of recursion.
Example 1
The predicate find_largest/2 takes a list of numbers as its first argument and
assigns the value of the largest element to its second argument (assumed to be an
unbound variable). It is assumed that the list contains at least one number.
find_largest([X|List],Maxval):find_biggest(List,Maxval,X).
find_biggest([],Currentlargest,Currentlargest).
find_biggest([A|L],Maxval,Currentlargest):A>Currentlargest,
find_biggest(L,Maxval,A).
find_biggest([A|L],Maxval,Currentlargest):A=<Currentlargest,
find_biggest(L,Maxval,Currentlargest).
Calling the find_largest goal with the list of numbers as its first argument
causes the first element of the list to be removed and passed to find_biggest as its
130 Logic Programming With Prolog
third argument (the largest number found so far). The remainder of the list is
passed to find_biggest as its first argument (a list of the numbers not yet
examined). The second argument of find_largest (Maxval) represents the overall
largest number and is unbound because the value is not yet known. It is passed to
find_biggest as its second argument.
The three arguments of find-biggest/3 represent in order:
•
the list of numbers not so far examined
•
the overall largest number (to be passed back to find_largest)
•
the largest number found so far.
So the first clause of find_biggest can be read as: 'if there are no more numbers
remaining unexamined, return the largest number so far (third argument) as the
value of the overall largest number (second argument)'. The second argument
becomes bound.
The second clause of find_biggest can be read as: 'if the list of numbers so far
not examined begins with value A and A is larger than the largest so far found, call
find_biggest (recursively) with L, the tail of the (first argument) list, as the new
first argument (list of unexamined numbers) and with A as the third argument
(largest number so far)'. The second argument (Maxval) is unbound.
The final clause of find_biggest can be read as: 'if the list of numbers so far not
examined begins with value A and A is not larger than the largest so far found, call
find_biggest (recursively) with L, the tail of the (first argument) list, as the new
first argument (list of unexamined numbers) and with the third argument (largest
number so far) unchanged'. The second argument (Maxval) is unbound.
?- find_largest([10,20,678,-4,-12,102,-5],M).
M = 678
?- find_largest([30,10],M).
M = 30
?- find_largest([234],M).
M = 234
Example 2
The front/2 predicate was defined in Section 9.6 as an example of the use of the
reverse built-in predicate. It takes a list as its first argument. If the second
argument is an unbound variable it is bound to a list which is the same as the first
list with the last element removed. For example if the first list is [a,b,c], the second
will be [a,b].
List Processing 131
The predicate can be defined more efficiently using recursion as follows.
front([X],[]).
front([X|Y],[X|Z]):-front(Y,Z).
The two clauses can be read as 'the front of a list with just one element is the
empty list' and 'the front of a list with head X and tail Y is the list with head X and
tail Z where Z is the front of Y', respectively.
?- front([alpha],L).
L = []
?- front([alpha,beta,gamma],LL).
LL = [alpha,beta]
?- front([[a,b],[c,d,e],[f,g,h]],L1).
L1 = [[a,b],[c,d,e]]
Example 3
One area in which different Prolog implementations can vary considerably is the
provision of built-in predicates for list processing. If your implementation does not
have member/2, reverse/2 or append/3 (described in Sections 9.4, 9.6 and 9.7
respectively) you can define your own with just a few clauses as shown below.
member(X,[X|L]).
member(X,[_|L]):-member(X,L).
reverse(L1,L2):-rev(L1,[],L2).
rev([],L,L).
rev([A|L],L1,L2):-rev(L,[A|L1],L2).
append([],L,L).
append([A|L1],L2,[A|L3]):-append(L1,L2,L3).
The two clauses defining member/2 just state that X is a member of any list
with head X (i.e. that begins with X) and that X is a member of any list for which it
is not the head if it is a member of the tail.
The definitions of the other two predicates are slightly more complex and are
left without explanation.
If your implementation of Prolog has any or all these predicates built-in,
attempting to load the above program will cause an error, as it is not permitted to
attempt to redefine a built-in predicate.
132 Logic Programming With Prolog
However the definitions can be tested by renaming the predicates
systematically as mymember, myreverse and myappend, say, giving the
following program.
mymember(X,[X|L]).
mymember(X,[_|L]):-mymember(X,L).
myreverse(L1,L2):-rev(L1,[],L2).
rev([],L,L).
rev([A|L],L1,L2):-rev(L,[A|L1],L2).
myappend([],L,L).
myappend([A|L1],L2,[A|L3]):-myappend(L1,L2,L3).
This can then be tested using some of the examples in Sections 9.4, 9.6 and 9.7,
for which it gives the same results in each case.
?- mymember(X,[a,b,c]).
X=a;
X=b;
X=c;
no
?- mymember(mypred(a,b,c),[q,r,s,mypred(a,b,c),w]).
yes
?- mymember(x,[]).
no
?- myreverse([1,2,3,4],L).
L = [4,3,2,1]
?- myreverse([[dog,cat],[1,2],[bird,mouse],[3,4,5,6]],L).
L = [[3,4,5,6],[bird,mouse],[1,2],[dog,cat]]
?- myappend([1,2,3,4],[5,6,7,8,9],L).
L = [1,2,3,4,5,6,7,8,9]
?- myappend([],[1,2,3],L).
L = [1,2,3]
List Processing 133
9.9 Using findall/3 to Create a List
It would often be desirable to find all the values that would satisfy a goal, not just
one of them. The findall/3 predicate provides a powerful facility for creating lists
of all such values. It is particularly useful when used in conjunction with the
Prolog database.
If the database contains the five clauses
person(john,smith,45,london).
person(mary,jones,28,edinburgh).
person(michael,wilson,62,bristol).
person(mark,smith,37,cardiff).
person(henry,roberts,23,london).
a list of all the surnames (the second argument of person) can be obtained using
findall by entering the goal:
?- findall(S,person(_,S,_,_),L).
This returns
L = [smith,jones,wilson,smith,roberts]
L is a list of all the values of variable S which satisfy the goal person(_,S,_,_).
The predicate findall/3 has three arguments. The first is generally an unbound
variable, but can be any term with at least one unbound variable as an argument (or
equivalently any list with at least one unbound variable as a list element).
The second argument must be a goal, i.e. must be in a form that could appear
on the right-hand side of a rule or be entered at the system prompt.
The third argument should be an unbound variable. Evaluating findall will
cause this to be bound to a list of all the possible values of the term (first argument)
that satisfy the goal (second argument).
More complex lists can be constructed by making the first argument a term
involving several variables, rather than using a single variable. For example
?- findall([Forename,Surname],person(Forename,Surname,_,_),L).
returns the list
L = [[john,smith],[mary,jones],[michael,wilson],[mark,smith],[henry,roberts]]
134 Logic Programming With Prolog
The term (first argument) can be embellished further, e.g.
?- findall([londoner,A,B],person(A,B,_,london),L).
returns the list
L = [[londoner,john,smith],[londoner,henry,roberts]]
Given a database containing clauses such as
age(john,45).
age(mary,28).
age(michael,62).
age(henry,23).
age(george,62).
age(bill,17).
age(martin,62).
the predicate oldest_list/1 defined below can be used to create a list of the names
of the oldest people in the database (in this case michael, george and martin, who
are all 62).
It begins by calling findall to find the ages of all the people in the database and
put them in a list Agelist.
It then uses the predicate find_largest (defined previously) to find the largest
of these values and bind variable Oldest to that value. Finally, it uses findall again
to create a list of the names of all the people of that age.
oldest_list(L):findall(A,age(_,A),Agelist),
find_largest(Agelist,Oldest),
findall(Name,age(Name,Oldest),L).
?- oldest_list(L).
L = [michael,george,martin]
The final example in this section shows a predicate find_under_30s used in
conjunction with the age predicate from the previous example to create a list of the
names of all those people under 30. This requires only one new clause.
find_under_30s(L):-findall(Name,(age(Name,A),A<30),L).
List Processing 135
?- find_under_30s(L).
L = [mary,henry,bill]
The second argument of findall is the goal (age(Name,A),A<30). It is
important to place parentheses around it so that it is treated as a single (compound)
goal with two component subgoals, not as two goals. Omitting the parentheses
would give a predicate named findall with four arguments. This would be an
entirely different predicate from findall/3 and one unknown to the Prolog system.
Chapter Summary
This chapter describes a flexible type of data object called a list. It shows how to
work through a list element by element from left to right using recursion to
perform the same or a similar operation on each element, how to manipulate lists
using built-in predicates and how to create a list containing all the possible values
that would satisfy a specified goal.
136 Logic Programming With Prolog
Practical Exercise 9
(1) Define and test a predicate pred1 that takes a list as its first argument and
returns the tail of the list as its second argument, e.g.
?- pred1([a,b,c],L).
L = [b,c]
(2) Define and test a predicate inc that takes a list of numbers as its first argument
and returns a list of the same numbers all increased by one as its second argument,
e.g.
?- inc([10,20,-7,0],L).
L = [11,21,-6,1]
(3) Define and test a predicate palindrome that checks whether a list reads the
same way forwards and backwards, e.g.
?- palindrome([a,b,c,b,a]).
yes
?- palindrome([a,b,c,d,e]).
no
(4) Define and test a predicate putfirst that adds a specified term to the beginning
of a list, e.g.
?- putfirst(a,[b,c,d,e],L).
L = [a,b,c,d,e]
(5) Define and test a predicate putlast that adds a specified term to the end of a list,
e.g.
?- putlast(e,[a,b,c,d],L).
L = [a,b,c,d,e]
(6) Using findall define and test predicates pred1/2, pred2/2 and pred3/2 that
modify a list, as shown in the following examples:
?- pred1([a,b,c,d,e],L).
L = [[a],[b],[c],[d],[e]]
?- pred2([a,b,c,d,e],L).
L = [pred(a,a),pred(b,b),pred(c,c),pred(d,d),pred(e,e)]
?- pred3([a,b,c,d,e],L).
L = [[element,a],[element,b],[element,c],[element,d],[element,e]]
10
String Processing
Chapter Aims
After reading this chapter you should be able to:
•
Use a built-in predicate to convert strings of characters to lists and vice
versa
•
Define predicates for the most common types of string processing.
10.1 Converting Strings of Characters To and From
Lists
An atom such as 'hello world' can be regarded as a string of characters. Prolog has
facilities to enable strings of this kind to be manipulated, e.g.
•
To join two strings such as 'Today is' and ' Tuesday' to form 'Today is
Tuesday'.
•
To remove initial spaces, e.g. to replace '
•
To find the part of the string after (or before) a specified string, e.g. to find
the name in a string such as 'My name is John Smith'.
hello world' by 'hello world'.
Prolog does this by converting strings to equivalent lists of numbers using the
name/2 predicate and then using list processing techniques such as those discussed
in Chapter 9, before (generally) converting the resulting lists back to strings.
The name/2 predicate takes two arguments. If the first is an atom and the
second is an unbound variable, evaluating a name goal will cause the variable to
138 Logic Programming With Prolog
be bound to a list of numbers equivalent to the string of characters that forms the
atom, for example:
?- name('Prolog Example',L).
L = [80,114,111,108,111,103,32,69,120,97,109,112,108,101]
As discussed in Chapter 5, each of the 256 possible characters has an
equivalent ASCII (American Standard Code for Information Interchange) value,
which is an integer from 0 to 255 inclusive.
The table of ASCII values corresponding to the most commonly used
characters is reproduced here for convenience.
9
tab
32
space
33
!
34
"
35
#
36
$
37
%
38
&
39
'
40
(
41
)
42
*
43
+
44
,
45
-
46
.
47
/
48-57 0 to 9
58
:
59
;
93
]
60
^
94
^
61
=
95
_
62
>
96
`
63
?
97-122 a to z
64
@
123
{
65-90 A to Z
124
|
91
[
125
}
92
\
126
~
In the example above, 80 is the ASCII value corresponding to the character P,
114 corresponds to r etc., so the list
[80,114,111,108,111,103,32,69,120,97,109,112,108,101]
corresponds to 'Prolog Example'.
The name predicate can also be used to perform the conversion in the other
direction, i.e. from a list of ASCII values to an equivalent atom.
?-name(A,[80,114,111,108,111,103,32,69,120,97,109,112,108,101]).
A = 'Prolog Example'
Once a string has been converted to list form it can be manipulated using any of
the facilities available for list processing to produce a new list or lists, which can
then be converted back to strings, again using the name predicate.
The examples in the following sections illustrate some of the programming
techniques involved in processing strings.
String Processing 139
10.2 Joining Two Strings
The predicate join2/3 defined below shows how to join two strings (the first two
arguments) to create a third string combining the two. The programming technique
used is a typical one in string processing: convert both strings to lists, concatenate
them using append and finally convert back the resulting list to a string.
/* Join two strings String1 and String2 to form a new
string Newstring */
join2(String1,String2,Newstring):name(String1,L1),name(String2,L2),
append(L1,L2,Newlist),
name(Newstring,Newlist).
?- join2('prolog',' example',S).
S = 'prolog example'
?- join2('','Prolog',S).
S = 'Prolog'
?- join2('Prolog','',S).
S = 'Prolog'
The predicate join3/4 defined below uses join2 twice to join three strings together.
/* Join three strings using the join2 predicate */
join3(String1,String2,String3,Newstring):join2(String1,String2,S),
join2(S,String3,Newstring).
?- join3('This is',' an',' example',Newstring).
Newstring = 'This is an example'
10.3 Trimming a String
This example shows how to define a predicate that will 'trim' a string, i.e. remove
any white space characters (spaces, tabs etc.) at the beginning or end. This will be
done in four stages.
140 Logic Programming With Prolog
Stage 1
Define and test a predicate trim/2 which takes a list of integers as its first argument
and an unbound variable as its second and binds the variable to the list with any
elements less than or equal to 32 at the left-hand end removed. This makes use of
the techniques described in Section 9.3, where the elements of a list are extracted
one by one using cons.
trim([A|L],L1):-A=<32,trim(L,L1).
trim([A|L],[A|L]):-A>32.
?- trim([26,32,17,45,18,27,94,18,16,9],X).
X = [45,18,27,94,18,16,9]
Stage 2
Define and test a predicate trim2/2 which takes a list of integers as its first
argument and an unbound variable as its second and binds the variable to the list
with any elements less than or equal to 32 at the right-hand end removed. This uses
trim in conjunction with the reverse predicate.
trim2(L,L1):reverse(L,Lrev),trim(Lrev,L2),reverse(L2,L1).
?- trim2([45,18,27,94,18,16,9],X).
X = [45,18,27,94]
Stage 3
Define and test a predicate trim3/2 which takes a list of integers as its first
argument and an unbound variable as its second and binds the variable to the list
with any elements less than or equal to 32 at the beginning and/or the end removed.
This uses trim to deal with the beginning of the list and trim2 to deal with the end.
trim3(L,L1):-trim(L,L2),trim2(L2,L1).
?- trim3([26,32,17,45,18,27,94,18,16,9],X).
X = [45,18,27,94]
Stage 4
Define and test a predicate trims/2 which takes an atom as its first argument and
an unbound variable as its second and binds the variable to the atom with any
white space characters at the beginning or end removed. Now that the list
String Processing 141
processing predicates trim, trim2 and trim3 have been defined, trims only needs
a one clause definition.
trims(S,Snew):-name(S,L),trim3(L,L1),name(Snew,L1).
The string used as the first argument below begins with three spaces and ends
with the combination 'tab, two spaces, tab'. All these are white space characters.
?- trims(' hello world
X = 'hello world'
',X).
10.4 Inputting a String of Characters
A very common requirement is to read an entire line of input either from the user's
terminal or from a text file. The Prolog built-in predicates for input are rather
limited. The read/1 predicate will only read a single term, terminated by a full
stop. The get0/1 and get/1 predicates will only read a single character.
The predicate readline/1 defined below takes an unbound variable as its
argument. Calling the predicate causes a line of input to be read from the user's
terminal and the variable to be bound to an atom comprising all the characters read
in, up to but not including a new line character (ASCII value 13).
Assume that the input is terminated by a character with ASCII value 13, which
is not included in the atom created.
readline(S):-readline1([],L),name(S,L),!.
readline1(Oldlist,L):-get0(X),process(Oldlist,X,L).
process(Oldlist,13,Oldlist).
process(Oldlist,X,L):append(Oldlist,[X],L1),readline1(L1,L).
?- readline(S).
: abcdefg
S = abcdefg
?- readline(S).
: this is an example ,.+-*/#@ - Note no quotes needed and no final full stop
S = ' this is an example ,.+-*/#@ - Note no quotes needed and no final full
stop'
The predicate readlineF/2 defined below is based on readline/1 but in an
adapted form, which deals with the slightly different requirements of a string of
characters read from a text file. In this book it is assumed that all such lines end in
two ASCII characters, with values 13 and 10, unlike lines of input from the user's
142 Logic Programming With Prolog
terminal which end with just one character (ASCII value 13). Once the character
with value 13 has been found, the first clause of predicate process ensures that the
character with value 10 is read and ignored.
/* readline adapted for input from text files */
readlineF(File,S):see(File),readline1([],L),name(S,L),!,seen.
readline1(Oldlist,L):-get0(X),process(Oldlist,X,L).
process(Oldlist,13,Oldlist):-get0(X).
/* To remove character with ASCII value 10 */
process(Oldlist,X,L):append(Oldlist,[X],L1),readline1(L1,L).
If text file file1.txt contains the following four lines
This is an example of
a text file with four lines each will be terminated by two invisible characters
with ASCII values 13 and 10
calling readfileF with first argument 'file1.txt' will cause the first line of the file to
be output.
?- readlineF('file1.txt',S).
S = 'This is an example of '
10.5 Searching a String
The predicate separate/3 defined below separates a list L into those elements
before and after the element 32.
separate(L,Before,After):append(Before,[32|After],L),!.
It does this using append with the second argument a list with head 32 –
another example of the value of using the cons notation to deconstruct a list.
?- separate([26,42,32,18,56,32,19,24],Before,After).
Before = [26,42] ,
After = [18,56,32,19,24]
String Processing 143
?- separate([32,24,86],Before,After).
Before = [] ,
After = [24,86]
?- separate([24,86,32],Before,After).
Before = [24,86] ,
After = []
?- separate([24,98,45,72],Before,After).
no
Predicate splitup/1 defined below starts by converting a string S to a list L, then
calls predicate findnext/1, which calls predicate proc/2 in a recursive fashion to
isolate the parts before and after each 32 element (corresponding to a space
character) in turn, convert it to a string and write it on a separate line.
separate(L,Before,After):append(Before,[32|After],L),!.
findnext(L):separate(L,Before,After),proc(Before,After).
findnext(L):-write('Last item is '),
name(S,L),write(S),nl.
proc(Before,After):-write('Next item is '),
name(S,Before),write(S),nl,findnext(After).
splitup(S):-name(S,L),findnext(L).
?- splitup('The time has come the walrus said').
Next item is The
Next item is time
Next item is has
Next item is come
Next item is the
Next item is walrus
Last item is said
yes
The predicate checkprolog/1, which takes an unbound variable as its argument,
causes a string of characters to be read from the user's terminal and the argument to
be bound to either the atom present or the atom absent, depending on whether or
not the input string includes the word Prolog.
The main predicate defined here is startList/2, which uses append to check
whether all the elements in list L1 appear at the beginning of list L2.
144 Logic Programming With Prolog
/* Uses predicate readline as defined previously */
startList(L1,L2):-append(L1,X,L2).
includedList(L1,[]):-!,fail.
includedList(L1,L2):-startList(L1,L2).
includedList(L1,[A|L2]):-includedList(L1,L2).
checkit(L,Plist,present):-includedList(Plist,L).
checkit(_,_,absent).
checkprolog(X):-readline(S),name(S,L),
name('Prolog',Plist),checkit(L,Plist,X),!.
?- checkprolog(X).
: Logic Programming with Prolog
X = present
?- checkprolog(X).
: Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto
X = absent
10.6 Dividing a String into Its Component Parts
The predicate splits/4 divides a string into the substrings to the left and right of
another string called a separator. Its arguments correspond to the string, the
separator, the left part and the right part in turn. For example:
?- splits('In the beginning was the word','the',Left,Right).
Left = 'In ' ,
Right = ' beginning was the word'
?- splits('my name is John Smith','is',Left,Right).
Left = 'my name ' ,
Right = ' John Smith'
The definition of this predicate is quite complex. There are four special cases.
(1) The separator appears more than once in the string. As can be seen from the
first example above, the first (i.e. 'leftmost') occurrence is taken.
(2) The string begins with the separator. In this case the left part should be set to
the separator and the right part should be set to the remainder of the string.
String Processing 145
(3) The string ends with the separator. In this case the right part should be set to the
separator and the left part should be set to the remainder of the string.
(4) The separator does not appear in the string. In this case the left part should be
set to the string and the right part should be set to ''.
The full definition is given below, followed by examples showing that cases
(2), (3) and (4) are dealt with correctly.
splits(S,Separator,Separator,R):name(Separator,L1),name(S,L3),
append(L1,L2,L3),name(R,L2),!.
splits(S,Separator,L,Separator):name(Separator,L2),name(S,L3),
append(L1,L2,L3),name(L,L1),!.
splits(S,Separator,Left,Right):name(S,L3),append(Lleft,Lrest,L3),
name(Separator,L4),append(L4,Lright,Lrest),
name(Left,Lleft),name(Right,Lright),!.
splits(S,_,S,''):-!.
?- splits('my name is John Smith','my name is ',Left,Right).
Left = 'my name is ' ,
Right = 'John Smith'
?- splits('my name is John Smith','John Smith',Left,Right).
Left = 'my name is ' ,
Right = 'John Smith'
?-splits('my name is my name is John Smith','is',Left,Right).
Left = 'my name ' ,
Right = ' my name is John Smith'
?- splits('my name is John Smith','Bill Smith',Left, Right).
Left = 'my name is John Smith' ,
Right = ''
Predicate remove_spaces/2 defined below uses predicate splits/4 to remove
any initial spaces from a string. The key idea is to split the string using a string
containing a space, i.e. ' ', as the separator. If the left part becomes bound to ' ' it
implies that case (2) above has occurred, i.e. the string must have begun with the
separator. In this case (first clause of remove2), the remainder of the string (the
146 Logic Programming With Prolog
right part) is used and any further spaces are removed from it in the same way. If
not, the final string is the same as the original string (second clause of remove2).
remove_spaces(S,S1):splits(S,' ',Sleft,Sright),
remove2(S,Sleft,Sright,S1),!.
remove2(S,' ',Sright,S1):-remove_spaces(Sright,S1).
remove2(S,_,_,S).
?- remove_spaces('hello world',X).
X = 'hello world'
?- remove_spaces(' hello world',X).
X = 'hello world'
Chapter Summary
This chapter describes the principal techniques of string processing in Prolog,
based on converting from atoms to lists and vice versa. The techniques are
illustrated with examples of user-defined predicates to meet common string
processing requirements.
String Processing 147
Practical Exercise 10
(1) Define and test a predicate spalindrome to check whether a string is a
palindrome, e.g.
?- spalindrome('abcd dcba').
yes
?- spalindrome('xyz').
no
(2) Use the name predicate to define and test a predicate remove_final that
removes any final spaces from a string, e.g.
?- remove_final('hello world ',X).
X = 'hello world'
?- remove_final('hello world',X).
X = 'hello world'
(3) Define and test a predicate replace to replace the first character in a string by
the character ? (which has ASCII value 63), e.g.
?- replace('abcde',X).
X = '?bcde'
11
More Advanced Features
Chapter Aims
After reading this chapter you should be able to:
•
Define operators to extend the basic Prolog language
•
Define basic operations on sets as well as numbers and strings
•
Use built-in predicates to manipulate terms.
Introduction
This chapter looks at some of the more advanced features provided by Prolog: the
use of operators to extend the language (e.g. to provide new arithmetic operators,
improved facilities for processing strings or facilities for processing sets) and
facilities for processing terms, converting them to lists or evaluating them as goals.
11.1 Extending Prolog: Arithmetic
Although Prolog allows the standard arithmetic operators (such as + - * and /) to be
used in arithmetic expressions, there is no similar convenient notation for
calculating factorials or to perform other less common but sometimes useful
operations such as adding the squares of two numbers.
The built-in predicate is/2 is used for evaluating arithmetic expressions. It is not
permitted for the Prolog programmer to redefine this by adding new operators (or
by any other means) and any attempt to do so would lead to a system error.
However, by using the technique described in this section the programmer can get
150 Logic Programming With Prolog
the same effect, so that expressions involving new arithmetic operators such as 6!
are permitted or even so that the definitions of standard operators such as + and are changed.
The key to this is to define a replacement for the is/2 operator. This new
operator will be called iss/2. There are two steps involved.
Step 1: Define iss/2 to be an operator
This can be done by entering the goal
?- op(710,xfx,iss).
at the system prompt or as a directive at the beginning of a program.
The third argument of the op/3 goal is of course the name of the operator. The
first argument is called its precedence. The second argument takes the rather
strange form xfx, denoting that iss is an infix operator, which takes two arguments
and will be written between them.
Step 2: Define the iss/2 operator
The simplest definition of iss/2 would be the single line
X iss Y:-X is Y.
This would simply make the operator iss equivalent to the built-in operator is,
so for example:
?- Z iss 6+sqrt(25)-2.
Z=9
This very simple definition would obviously not achieve much. An improved
(but not yet ideal) attempt at defining iss is as follows. The effect of using iss in
combination with the different types of arithmetic operator (+ - * / etc.) is specified
explicitly in the first eight clauses. All other cases (e.g. expressions involving sqrt,
sin etc.) are dealt with by the final clause. With these definitions the operator iss
still has the same effect as is.
Y iss A+B:-Y is A+B,!.
Y iss A-B:-Y is A-B,!.
Y iss A*B:-Y is A*B,!.
Y iss A/B:-Y is A/B,!.
Y iss A//B:-Y is A//B,!.
Y iss A^B:-Y is A^B,!.
Y iss +A:-Y is A,!.
More Advanced Features 151
Y iss -A:- Y is -A,!.
Y iss X:- Y is X,!.
?- Y iss 6+4*3-2.
Y = 16
?- X iss 3,Y iss X+5.6-3*10+100.5.
X=3,
Y = 79.1
?-Y iss (8+4)/3-(6*7).
Y = -38
?- A=3,B=4,Y iss sqrt(A*A+B*B).
A=3,
B=4,
Y=5
?- Y iss 6+sqrt(25).
Y = 11
?- Y iss 6+sqrt(10+15).
Y = 11
Defining ! as a Factorial Operator
Starting from this more elaborate definition of iss, we can now add further
operators.
The mathematical function factorial is defined only for integer arguments. The
value of 'factorial 6' is 6 x 5 x 4 x 3 x 2 x 1 and is written as 6! (i.e. the number 6
followed by an exclamation mark).
In general the value of N! is Nx(N-1)!. This leads to a two-line recursive
definition of a predicate factorial/2:
factorial(1,1):-!.
factorial(N,Y):-N1 is N-1,factorial(N1,Y1),
Y is N*Y1.
It is assumed that the first argument will always be an integer or a variable
bound to an integer and the second argument is an unbound variable. Then, for
example, the product 6 x 5 x 4 x 3 x 2 x 1 can be found by
?- factorial(6,Y).
Y = 720
152 Logic Programming With Prolog
This predicate can now be used to define a new arithmetic operator ! which will
enable terms such as 6! or N! to be written when evaluating an arithmetic
expression using the iss predicate. As usual, there are two actions required to do
this.
Step 1: Define ! to be an operator
This can be done by entering the goal
?- op(200,xf,!).
The atom xf denotes that ! is a postfix operator, which will appear after its
argument, e.g. 6!. Its precedence is 200.
Step 2: Define the ! predicate
Using the definition of the factorial predicate already given, the ! operator (or
rather its effect when used together with the iss operator) can be defined by adding
the following clause to the definition of iss, say as the first line.
Y iss N!:-N1 iss N,factorial(N1,Y),!.
This allows the exclamation mark character to be used in a convenient way to
represent factorials.
?- Y iss 6!.
Y = 720
?- Y iss (3+2)!.
Y = 120
However, there is a flaw in the definition of iss. Entering a goal such as
?- Y iss 5!+6!.
will cause Prolog to crash with an error message such as 'Function Not Defined'.
The reason is that to evaluate this expression, Prolog makes use of the
definition of the + operator, which is
Y iss A+B:-Y is A+B,!.
This causes it to try to evaluate the goal
Y is 5!+6!
More Advanced Features 153
which causes an error as in this context 5! and 6! are not numbers. They have no
meaning at all outside their definition for the iss predicate.
The most satisfactory way of dealing with this problem is to modify the
definition of the iss operator so that its arguments are themselves evaluated using
iss before adding, multiplying etc. their values. This requires every clause in the
definition of iss/2 to be modified, expect for the last, and gives the following
revised program.
?- op(710,xfx,iss).
?- op(200,xf,!).
factorial(1,1):-!.
factorial(N,Y):-N1 is N-1,factorial(N1,Y1),Y is N*Y1.
Y iss N!:-N1 iss N,factorial(N1,Y),!.
Y iss A+B :-A1 iss A,B1 iss B,Y is A1+B1,!.
Y iss A-B :-A1 iss A,B1 iss B,Y is A1-B1,!.
Y iss A*B :-A1 iss A,B1 iss B,Y is A1*B1,!.
Y iss A/B :-A1 iss A,B1 iss B,Y is A1/B1,!.
Y iss A//B :-A1 iss A,B1 iss B,Y is A1//B1,!.
Y iss A^B :-A1 iss A,B1 iss B,Y is A1^B1,!.
Y iss +A :-Y iss A,!.
Y iss -A :- A1 iss A,Y is -A1,!.
Y iss X :- Y is X,!.
With the new definition of the + operator, if either of its arguments is an
expression such as 5! it is converted to a number before it is used. If an argument is
a number, it is 'converted' to itself by the final clause.
The ! operator now works as expected. When the goal Y iss 5!+6! is evaluated,
the system first applies iss to 5! and to 6! producing the numbers 120 and 720,
respectively, and then adds them together.
Y iss 6!.
Y = 720
Y iss (3+2)!.
Y = 120
?- Y iss 5!+6!.
Y = 840
154 Logic Programming With Prolog
?- Y iss 4+2,Z iss Y!+3!-4!.
Y=6,
Z = 702
?- Y iss (3!)!.
Y = 720
?- Y iss -(3!).
Y = -6
Note that the above definition of iss is still not watertight. Expressions such as
sqrt(3!) will cause the system to crash. This can be overcome by adding additional
clauses such as
Y iss sqrt(A):-A1 iss A, Y is sqrt(A1),!.
for all the arithmetic functions, such as sqrt, tan and sin with which it is intended
to use the new operator.
Defining ** as a Sum of Squares Operator
As well as factorial, we can define new operators that perform any operations we
wish. For example we might want to have an infix operator ** that returns the sum
of the squares of its two arguments. This can be defined as follows.
Step 1: Define ** to be an operator
This can be done by entering the goal
?- op(200,yfx,**).
This specifies that ** is an infix operator, which will appear between its two
arguments, e.g. 3**4. Its precedence is 200.
Step 2: Define the ** operator
The ** operator can be defined by adding the following clause to the definition of
iss, anywhere except the final line (which is a 'catch all').
Y iss A**B:- A1 iss A, B1 iss B, Y is A1*A1+B1*B1,!.
?- Y iss 3**2.
Y = 13
?- Y iss (3**2)+2.
Y = 15
More Advanced Features 155
?- Y iss 6+3**4+8+1**2-10.
Y = 34
?- Y iss (3**1)**(2**1).
Y = 125
Redefining Addition and Subtraction
Now that we have the iss predicate we can even use it to 'redefine' addition and
subtraction if we wish. If we change the following clauses in the definition of iss
Y iss A+B:- A1 iss A,B1 iss B,Y is A1+B1,!.
Y iss A-B:- A1 iss A,B1 iss B,Y is A1-B1,!.
to
Y iss A+B:- A1 iss A,B1 iss B,Y is A1-B1,!.
Y iss A-B:- A1 iss A,B1 iss B,Y is A1+B1,!.
The effect will be to cause + to subtract and - to add. For example
?- Y iss 6+4.
Y=2
?- Y iss 6-4.
Y = 10
11.2 Extending Prolog: Operations on Strings
Now that we have this useful predicate iss/2 we do not have to restrict its use to
numbers. The join2/3 predicate was defined in Chapter 10. It takes three
arguments, the first two of which are atoms (or variables bound to atoms). These
atoms are regarded as strings.
Assuming the third argument is an unbound variable, evaluating a join2 goal
will cause it to be bound to another atom, which is the first two joined together.
For reference, the definition of the join2 predicate and some examples of its
use are repeated below.
/* Join two strings String1 and String2 to form a new
string Newstring */
join2(String1,String2,Newstring):name(String1,L1),name(String2,L2),
156 Logic Programming With Prolog
append(L1,L2,Newlist),
name(Newstring,Newlist).
?- join2('prolog',' example',S).
S = 'prolog example'
?- join2('','Prolog',S).
S = 'Prolog'
?- join2('Prolog','',S).
S = 'Prolog'
Although this is fine as far as it goes, it would be more convenient to be able to
use an operator such as ++ to join two strings. This can be achieved by first
defining the operator, by entering a goal such as
?- op(200,yfx,++).
and then adding the following clause to the definition of the iss predicate:
S iss S1++S2:-join2(S1,S2,S),!.
?- S iss 'hello'++' world'.
S = 'hello world'
Joining more than two strings together using ++ encounters problems similar to
those in the last section. Evaluating the sequence of goals
?- X='United States', Y='of America',Z iss X++' '++Y.
will cause the system to crash. The join2 predicate cannot deal with an argument
such as X++' '.
As for factorial, we need to process the arguments on either side of ++ to
ensure that they are strings (atoms), not expressions, before they are used. To do
this we define a predicate convert/2. If the first argument is an atom, it binds the
second argument to that value. If not, it must be an expression using ++, such as
'hello'++'world'. In this case iss is used to calculate the new string and this is
returned as the second argument.
The built-in predicate atom/1 tests whether or not a term is an atom. The goal
atom(X) succeeds if and only if X is an atom or a variable bound to an atom.
Using this we can define the convert/2 predicate as follows:
convert(X,X):-atom(X).
convert(X,X1):-X1 iss X.
More Advanced Features 157
The previous definition of ++ can now be replaced by the clause:
S iss S1++S2:convert(S1,A),convert(S2,B),join2(A,B,S),!.
With this new definition, ++ can be used to join any number of strings together.
?- S iss 'hello'++' world'.
S = 'hello world'
?- X='United States', Y='of America',Z iss X++' '++Y.
X = 'United States' ,
Y = 'of America' ,
Z = 'United States of America'
?- X='United States', Y='of America',Z iss 'This is the '++X++' '++Y.
X = 'United States' ,
Y = 'of America' ,
Z = 'This is the United States of America'
11.3 Extending Prolog: Sets
An important branch of mathematics is known as set theory. A detailed description
of this is outside the scope of this book, but essentially a set is similar to a list of
atoms, such as [a,b,c,d] but with two major differences: the order of the elements
is of no significance and no element may occur twice in a set.
Given two sets X and Y, a standard requirement is to produce any or all of three
new sets based on the first two:
•
the intersection – those elements that are in both sets
•
the union – those elements that are in either set (or both)
•
the difference – those elements that are in the first set but not the second.
We will denote these by the expressions X and Y, X or Y and X-Y,
respectively.
To define these set operations, we will start by defining a new predicate sis/2,
in the same way as iss/2 was defined previously, i.e. by entering a goal such as
?- op(710,xfx,sis).
We now need to define the and and or operators for intersection and union,
which we can do by entering goals such as:
158 Logic Programming With Prolog
?- op(200,yfx,and).
?-op(200,yfx,or).
There is no need to define the minus operator for difference, as it is already
defined.
We now need to define the meaning of and, or and - when used in conjunction
with sis. We do this using the predicate findall/3:
Y sis A and B :findall(X,(member(X,A),member(X,B)),Y),!.
Y sis A or B :findall(X,(member(X,A);member(X,B)),Y),!.
Y sis A-B :findall(X,(member(X,A),not(member(X,B))),Y),!.
corresponding to find all elements that are members of both A and B, members of
either A or B (or both) and members of A but not B, respectively. Unfortunately
the second clause is not quite correct.
?- X=[a,b,c,d],Y=[e,b,f,c,g],A sis X and Y.
X = [a,b,c,d] ,
Y = [e,b,f,c,g] ,
A = [b,c]
?- X=[a,b,c,d],Y=[e,b,f,c,g],A sis X or Y.
X = [a,b,c,d] ,
Y = [e,b,f,c,g] ,
A = [a,b,c,d,e,b,f,c,g]
?- X=[a,b,c,d],Y=[e,b,f,c,g],A sis X-Y.
X = [a,b,c,d] ,
Y = [e,b,f,c,g] ,
A = [a,d]
The goal A sis X or Y causes variable A to be bound to a list containing all
those elements that are in either X or Y. However elements b and c occur in both X
and Y, and so appear twice in list A, which is therefore not a valid set.
To get around this, we can change the definition of the or operator to:
Y sis A or B:findall(X,(member(X,A);(member(X,B),not(member(X,A)))),Y),!.
More Advanced Features 159
corresponding to: find all elements X that are members of A or are members of B
but not members of A.
With this new definition, the or operator gives the expected result.
?- X=[a,b,c,d],Y=[e,b,f,c,g],A sis X or Y.
X = [a,b,c,d] ,
Y = [e,b,f,c,g] ,
A = [a,b,c,d,e,f,g]
If we want to combine several operators in a single expression, e.g. X and Y
and Z, we need to change the definitions of and, or and - when used with sis, to
allow for the possibility of an argument being an expression not a list.
We can do this by first applying sis to each argument, replacing it by a list if it
is an expression, or by itself if it is a list. This requires an additional final clause to
be added, specifying that applying sis to 'anything else' (i.e. a list) gives the same
value.
This gives a revised definition of sis/2 as follows:
Y sis A and B :A1 sis A,B1 sis B,
findall(X,(member(X,A1),member(X,B1)),Y),!.
Y sis A or B:A1 sis A,B1 sis B,
findall(X,(member(X,A1);(member(X,B1),not(member(X,A1)))),Y),!.
Y sis A-B :A1 sis A,B1 sis B,
findall(X,(member(X,A1),not(member(X,B1))),Y),!.
A sis A:-!.
?- X=[a,b,c,d],Y=[e,b,f,c,g],Z=[a,e,c,g,h],A sis X and Y and Z.
X = [a,b,c,d] ,
Y = [e,b,f,c,g] ,
Z = [a,e,c,g,h] ,
A = [c]
?- X=[a,b,c,d],Y=[e,b,f,c,g],Z=[a,e,c,g,h],A sis X or Y or Z.
X = [a,b,c,d] ,
Y = [e,b,f,c,g] ,
Z = [a,e,c,g,h] ,
A = [a,b,c,d,e,f,g,h]
160 Logic Programming With Prolog
?- X=[a,b,c,d],Y=[e,b,f,c,g],Z=[a,e,c,g,h],A sis X-Y-Z.
X = [a,b,c,d] ,
Y = [e,b,f,c,g] ,
Z = [a,e,c,g,h] ,
A = [d]
11.4 Processing Terms
Prolog has several facilities for processing terms, which can be useful for more
advanced applications. Terms can be decomposed into their functor and arity, a
specified argument can be extracted from a compound term, terms can be
converted to lists or vice versa and a term can be evaluated as a goal.
Using the univ Operator to Convert Lists to Terms
The built-in infix operator =.. is known (for obscure historical reasons) as 'univ'.
The operator is written as an = sign followed by two full stops, i.e. three characters.
Evaluating the goal
X=..[member,A,L]
causes variable X to be bound to the term member(A,L).
Evaluating the goal
X=..[colour,red]
causes X to be bound to the term colour(red). X can then be used just like any
other term, so for example
?-X=..[colour,red],assertz(X).
would place the clause colour(red) in the database.
?- X=..[colour,red],assertz(X).
X = colour(red)
?- colour(red).
yes
It is also possible to go the other way. If the first argument of 'univ' is a term
and the second is an unbound variable, the latter is bound to a list that corresponds
to the term, with the first element the functor and the remaining elements the
arguments of the compound term in order. For example,
?- data(6,green,mypred(26,blue))=..L.
More Advanced Features 161
will bind the variable L to a list
L = [data,6,green,mypred(26,blue)]
The call/1 Predicate
The call/1 predicate takes a single argument, which must be a call term, i.e. an
atom or a compound term, or a variable bound to a call term. The term is evaluated
as if it were a goal.
?- call(write('hello world')).
hello worldyes
?- X=write('hello world'),call(X).
hello worldX = write('hello world')
call/1 can be used to evaluate more than one goal if they are separated by
commas and enclosed in parentheses.
?- call((write('hello world'),nl)).
hello world
yes
?- X=(write('hello world'),nl),call(X).
hello world
X = (write('hello world'),nl)
In the above cases there is no benefit over entering the goal or goals directly.
However, if the value of X is not known in advance but is a calculated value or is
read from a file, being able to call X as a goal can be very useful.
The call predicate is sometimes used in conjunction with univ, for example:
?- X=..[write,'hello world'],call(X).
hello worldX = write('hello world')
If the database contains the clause
greet(Z):-write('Hello '),write(Z),nl,
write('How are you?'),nl.
we can cause a greet goal to be evaluated by:
?- X=..[greet,martin],call(X).
Hello martin
How are you?
X = greet(martin)
162 Logic Programming With Prolog
The functor/3 Predicate
The built-in predicate functor/3 takes three arguments. If the first argument is an
atom or a compound term or a variable bound to one of those, and the other two
arguments are unbound, the second argument will be bound to the functor of the
first argument and the third will be bound to its arity. For this purpose an atom is
considered to have arity zero.
?- functor(write('hello world'),A,B).
A = write ,
B=1
?- functor(city(london,england,europe),Funct,Ar).
Funct = city ,
Ar = 3
?- Z=person(a,b,c,d),functor(Z,F,A).
Z = person(a,b,c,d) ,
F = person ,
A=4
?- functor(start,F,A).
F = start ,
A=0
?- functor(a+b,F,A).
F=+,
A=2
If the first argument is an unbound variable, the second is an atom, and the third
is a positive integer, the variable is bound to a compound term with the given
functor and arity, with all its arguments unbound variables. If the third argument is
zero, the first argument is bound to an atom.
?- functor(T,person,4).
T = person(_42952,_42954,_42956,_42958)
?- functor(T,start,0).
T = start
The arg/3 Predicate
The built-in predicate arg/3 can be used to find a specified argument of a
compound term. The first two arguments must be bound to a positive integer and a
compound term, respectively. If the third argument is an unbound variable, it is
bound to the value of the specified argument of the compound term.
More Advanced Features 163
?- arg(3,person(mary,jones,doctor,london),X).
X = doctor
?- N=2,T=person(mary,jones,doctor,london),arg(N,T,X).
N=2,
T = person(mary,jones,doctor,london) ,
X = jones
We can use functor/3 and arg/3 to define a predicate to determine whether or
not two call terms will unify, as described in Section 3.1. The built-in operator =/2
can be used for this, but we shall assume that it is not available. In summary, an
atom can only be unified with the same atom and a compound term can only be
unified with a compound term with the same functor and arity. Two compound
terms with the same functor and arity unify if and only if their arguments unify
pairwise. A first version of a unify/2 predicate embodying these rules is as follows.
unify(CT1,CT2):functor(CT1,Funct1,Arity1),
functor(CT2,Funct2,Arity2),
compare(Funct1,Arity1,Funct2,Arity2).
compare(F,0,F,0).
/* Same atom*/
compare(F,0,F1,0):-fail.
/* Will not unify - different atoms */
compare(F,A,F,A):-write('May unify - check whether
arguments unify pairwise'),nl.
compare(_,_,_,_):-fail.
?- unify(person(a,b,c,d),person(a,b,c)).
no
?- unify(person(a,b,c,d),person2(a,b,c,d)).
no
?- unify(london,london).
yes
?- unify(london,washington).
no
?- unify(dog,person(a,b)).
no
164 Logic Programming With Prolog
?- unify(person(a,b,c,d),person(a,b,c,f)).
May unify - check whether arguments unify pairwise
yes
To extend this program we need to modify the penultimate compare clause.
We start by adding two additional arguments to predicate compare, so we can pass
the two call terms to it from unify. We then change the penultimate clause of
compare so that when the call terms are both compound terms with the same
functor and arity they are passed to a new predicate unify2, which examines their
arguments pairwise.
unify2 works by converting the two compound terms to lists using univ, then
removing the heads of the lists (the common functor) and passing them to predicate
paircheck/2, which succeeds if and only if the corresponding pairs of list elements
can be unified. The standard Prolog unification is used to check that each pair of
elements can be unified.
unify(CT1,CT2):functor(CT1,Funct1,Arity1),
functor(CT2,Funct2,Arity2),
compare(CT1,CT2,Funct1,Arity1,Funct2,Arity2).
compare(CT1,CT2,F,0,F,0).
/* Same atom*/
compare(CT1,CT2,F,0,F1,0):-fail. /* Different atoms */
compare(CT1,CT2,F,A,F,A):-unify2(CT1,CT2),!.
compare(CT1,CT2,_,_,_,_):-fail.
unify2(CT1,CT2):CT1=..[F|L1],CT2=..[F|L2],!,paircheck(L1,L2).
paircheck([],[]).
paircheck([A|L1],[A|L2]):-paircheck(L1,L2).
?- unify(person(a,b,c,d),person(a,b,X,Y)).
X=c,
Y=d
?- unify(pred(6,A,[X,Y,[a,b,c]],pred2(p,q)),pred(P1,Q1,L,Z)).
A = Q1 = _ ,
X=_,
Y=_,
P1 = 6 ,
L = [X,Y,[a,b,c]] ,
Z = pred2(p,q)
More Advanced Features 165
?- unify(person(a,b,c,d),person(a,b,X,X)).
no
Note that this definition of unify is a simplified one that does not ensure that
there are no common variables in the two compound terms. This needs to be dealt
with separately.
Chapter Summary
This chapter describes some of the more advanced features provided by Prolog: the
use of operators to extend the language (e.g. to provide new arithmetic operators,
improved facilities for processing strings or facilities for processing sets) and
facilities for processing terms, including finding their functors and arities,
extracting a specified argument of a term, converting terms to lists or evaluating
terms as goals. A user-defined predicate to unify two compound terms is given.
166 Logic Programming With Prolog
Practical Exercise 11
(1) Extend the definition of the iss/2 predicate with prefix operators for the head
and tail of a list that can be used in the following way:
?- Y iss head [a,b,c].
Y=a
?- Y iss tail [a,b,c].
Y = [b,c]
(2) Define and test a predicate addArg/3. Its first argument must be a compound
term or a variable bound to one. The second argument must be a term. The third
argument must be an unbound variable. Evaluating the goal should bind this
variable to a compound term that is the same as the original one, with the specified
term added as an additional final argument. Your definition should work however
many arguments the compound term used for the first argument has. For example,
it should produce the following output:
?- addArg(person(john,smith,25),london,T).
T = person(john,smith,25,london)
Appendix 1
Built-in Predicates
This appendix gives a brief description of each built-in predicate mentioned in this
book and some others. They are all 'standard' predicates, which should be available
in every version of Prolog, but inevitably there may be some exceptions to this. It
is also possible that in some implementations of Prolog the definitions may vary
slightly from those given here. In cases of disagreement, the definitions given in
the supplier's documentation should always be taken as definitive.
Name of Predicate !/0 [exclamation mark symbol, pronounced 'cut']
Syntax !
Description
Always succeeds. Used to control backtracking (see Chapter 7).
Name of Predicate append/3
Syntax append(First,Second,Whole )
Description
Join or split lists (see Chapter 9).
168 Logic Programming With Prolog
Name of Predicate arg/3
Syntax arg(N,Term,Arg)
Description
N must be a positive integer and Term must be a compound term. Arg is unified
with the Nth argument of Term.
Name of Predicate asserta/1
Syntax asserta(Clause)
Description
Adds a clause to the definition of a predicate at the beginning of its sequence of
existing clauses (if any).
Name of Predicate assertz/1
Syntax assertz(Clause)
Description
Adds a clause to the definition of a predicate at the end of its sequence of existing
clauses (if any).
Name of Predicate atom/1
Syntax atom(Term)
Description
Succeeds if and only if the given Term is a Prolog atom.
Built-in Predicates 169
Name of Predicate atomic/1
Syntax atomic(Term)
Description
Succeeds if and only if Term is an atom, a number, or a variable bound to either.
Name of Predicate call/1
Syntax call(Goal)
Description
Calls the given Goal. Succeeds if Goal succeeds and fails if Goal fails.
Name of Predicate consult/1
Syntax consult(Filename)
Description
Loads the program contained in the named disk file.
Name of Predicate dynamic/1
Syntax dynamic(predicate_specification)
Description
Used to specify that a predicate is 'dynamic', i.e. may be modified (see Page 110).
Name of Predicate fail/0
Syntax fail
Description
Always fails. Used to force a program to backtrack.
170 Logic Programming With Prolog
Name of Predicate findall/3
Syntax findall(Term,Goal,List)
Description
Returns a list of all the instances of Term that satisfy goal Goal.
Name of Predicate functor/3
Syntax functor(Term,Functor,Arity)
Description
Succeeds if Term has the specified Functor and Arity.
Name of Predicate get/1
Syntax get(Char)
Description
This reads the next 'printable' (i.e. non-white-space) character from the current
input stream, and unifies Char with its integer character code.
Name of Predicate get0/1
Syntax get0(Char)
Description
This reads the next character from the current input stream, and unifies Char with
its integer character code.
Built-in Predicates 171
Name of Predicate halt/0
Syntax halt
Description
Terminates the current Prolog session and exits to the operating system.
Name of Predicate integer/1
Syntax integer(Term)
Description
Succeeds if and only if Term is an integer.
Name of Predicate length/2
Syntax length(List,Length)
Description
Tests the length of a list (see Chapter 9).
Name of Predicate listing/1
Syntax listing(Atom)
Description
Lists all predicates with the given name, irrespective of their arity.
172 Logic Programming With Prolog
Name of Predicate member/2
Syntax member(Term,List )
Description
Gets or checks a member of a list (see Chapter 9).
Name of Predicate name/2
Syntax name(Atom,List)
Description
Converts between an atom and a list of characters (see Chapter 10).
Name of Predicate nl/0
Syntax nl
Description
Outputs a carriage return and line feed to the current output stream, to complete a
line of output.
Name of Predicate op/3
Syntax op(Precedence,Type,Name)
Description
Used to set, reset or clear the definition of an operator, using the given Precedence,
Type and Name.
Built-in Predicates 173
Name of Predicate put/1
Syntax put(Integer)
Description
Outputs the character corresponding to Integer to the current output stream.
Name of Predicate read/1
Syntax read(Var)
Description
Reads a term from the current input stream and attempts to assign the value to Var,
which should previously be unbound.
Name of Predicate reconsult/1
Syntax reconsult(Filename)
Description
Loads the program contained in the named disk file. Any clauses for the same
predicates already in the database are first deleted.
Name of Predicate repeat/0
Syntax repeat
Description
Always succeeds when called, both when called and on backtracking. Used to
provide a looping facility.
174 Logic Programming With Prolog
Name of Predicate retract/1
Syntax retract(Clause)
Description
Deletes the first matching clause from a predicate (see Chapter 8).
Name of Predicate retractall/1
Syntax retractall(Head)
Description
Deletes all clauses whose heads match the given Head (see Chapter 8).
Name of Predicate reverse/2
Syntax reverse(List,Reverse)
Description
Reverses the order of elements in a list (see Chapter 9).
Name of Predicate see/1
Syntax see(Stream)
Description
Sets Stream to be the current input stream. Stream may be the name of a disk file
or the atom user (referring to the console input device). If Stream refers to a disk
file which is not open, it is automatically opened for reading. If the file is already
open, input continues from the point immediately after the previously-read
character.
Built-in Predicates 175
Name of Predicate seeing/1
Syntax seeing(Stream)
Description
Stream is bound to the name of the current input stream, which may be a disk file
or the atom user (referring to the console input device).
Name of Predicate seen/0
Syntax seen
Description
Closes the file associated with the current input stream, and resets the current input
stream to user.
Name of Predicate statistics/0
Syntax statistics
Description
Displays statistics about the current session.
Name of Predicate tell/1
Syntax tell(Stream)
Description
Sets the current output stream, which may be a disk file or the atom user (referring
to the console output device). If Stream refers to a disk file which is not yet open, it
is automatically opened for writing. If the file was already open, output continues
from the point immediately after the previously written character.
176 Logic Programming With Prolog
Name of Predicate telling/1
Syntax telling(Stream)
Description
Gets the current output Stream, which may be a disk file or the atom user (referring
to the console output device). Stream is bound to the name of the current output
stream.
Name of Predicate told/0
Syntax told
Description
Closes the file associated with the current output stream, and resets the current
output stream to user.
Name of Predicate true/0
Syntax true
Description
Always succeeds.
Name of Predicate write/1
Syntax write(Term)
Description
Writes Term to the current output stream, in unquoted syntax.
Built-in Predicates 177
Name of Predicate writeq/1
Syntax writeq(Term)
Description
Writes Term to the current output stream, in quoted syntax.
Appendix 2
Built-in Operators
This appendix gives a brief description of each built-in operator mentioned in this
book and some others. They are all 'standard' operators, which should be available
in every version of Prolog, but inevitably there may be some exceptions to this. It
is also possible that in some implementations of Prolog the definitions may vary
slightly from those given here. In cases of disagreement, the definitions given in
the supplier's documentation should always be taken as definitive.
Name of Operator , [comma]
Type of Operator infix
Syntax Goal1,Goal2
Description
Succeeds if and only if Goal1 and Goal2 are both true.
Name of Operator ; [semicolon]
Type of Operator infix
Syntax Goal1;Goal2
Description
Succeeds if either Goal1 or Goal2 is true (or both).
180 Logic Programming With Prolog
Name of Operator =
Type of Operator infix
Syntax Term1 = Term2
Description
Succeeds if terms Term1 and Term2 unify (see Chapter 4).
Name of Operator \=
Type of Operator infix
Syntax Term\=Term2
Description
Succeeds if Term1 does not unify with Term2 (see Chapter 4).
Name of Operator ==
Type of Operator infix
Syntax Term1 == Term2
Description
Succeeds if Term1 is identical to Term2 (see Chapter 4).
Name of Operator \==
Type of Operator infix
Syntax Term1 \== Term2
Description
Succeeds if Term1 is not identical to Term2 (see Chapter 4).
Built-in Operators 181
Name of Operator =:=
Type of Operator infix
Syntax Exp1 =:= Exp2
Description
Succeeds if the arithmetic expressions Exp1 and Exp2 evaluate to the same
numerical value (see Chapter 4).
Name of Operator =\=
Type of Operator infix
Syntax Exp1 =\= Exp2
Description
Succeeds if the arithmetic expressions Exp1 and Exp2 do not evaluate to the same
numerical value (see Chapter 4).
Name of Operator =.. [pronounced 'univ']
Type of Operator infix
Syntax Term=..List
Description
Converts from a list to a term or vice versa (see Chapter 11).
182 Logic Programming With Prolog
Name of Operator <
Type of Operator infix
Syntax Exp1<Exp2
Description
Succeeds if the value of arithmetic expression Exp1 is less than the value of
arithmetic expression Exp2.
Name of Operator =<
Type of Operator infix
Syntax Exp1=<Exp2
Description
Succeeds if the value of arithmetic expression Exp1 is less than or equal to the
value of arithmetic expression Exp2.
Name of Operator >
Type of Operator infix
Syntax Exp1>Exp2
Description
Succeeds if the value of arithmetic expression Exp1 is greater than the value of
arithmetic expression Exp2.
Built-in Operators 183
Name of Operator >=
Type of Operator infix
Syntax Exp1>=Exp2
Description
Succeeds if the value of arithmetic expression Exp1 is greater than or equal to the
value of arithmetic expression Exp2.
Name of Operator is/2
Type of Operator infix
Syntax Result is Expression
Description
Expression must be a valid arithmetic expression which is evaluated to give a
number. If Result is an unbound variable (the usual case) the variable is bound to
the value of the expression. If Result is a bound variable with a numerical value or
a number, the goal succeeds if the values of both sides of the is operator are the
same and fails otherwise.
Name of Operator not/1
Type of Operator prefix
Syntax not Goal
Description
Succeeds if Goal fails, fails if Goal succeeds.
Appendix 3
Specimen Solutions to Practical Exercises
Practical Exercise 1
Question 2
A suitable program would comprise the five clauses:
animal(lion).
animal(tiger).
animal(cow).
carnivore(lion).
carnivore(tiger).
Suitable sequences of goals to test (a) to (d) are as follows.
?- animal(tiger).
yes
?- animal(cow),animal(tiger).
yes
?- animal(lion),carnivore(lion).
yes
?- animal(cow),carnivore(cow).
no
186 Logic Programming With Prolog
Question 3
Here is the output produced by one Prolog system for the specified goals, with
explanations inserted in italic.
?- write(hello).
helloyes
?- write(Hello).
_12544Hello = _
This obscure output is produced because Hello is not an atom (it begins with a
capital letter). To print Hello World with a capital H it must be enclosed in quotes.
?- write('Hello!').
Hello!yes
It is not (usually) possible to suppress the final 'yes', which indicates that the goal
has ben satisfied. However the output can be made more readable by using 'nl' to
generate a newline after 'Hello!'.
?- write('Hello!'),nl.
Hello!
yes
?- 100=100.
yes
?- 100=1000/10.
no
Using = is not the right way to do arithmetic – see Chapter 4.
?- 100 is 1000/10.
yes
?- 1000 is 100*10.
yes
?- 2 is (5+7)/6.
yes
?- 74 is (5+7)*6.
no
Specimen Solutions to Practical Exercises 187
Practical Exercise 2
Question 1
A suitable series of goals is as follows:
?- animal(mammal,A,_,_).
A = tiger ;
A = hyena ;
A = lion ;
A = zebra
?- animal(mammal,A,carnivore,_).
A = tiger ;
A = hyena ;
A = lion ;
no
?- animal(mammal,A,_,stripes).
A = tiger ;
A = zebra
?- animal(reptile,A,_,mane).
no
Question 2
A suitable additional rule would be
couple(X,Y):-person(X,male),person(Y,female).
Testing this gives the following output:
?- couple(X,Y).
X = bill ,
Y = carol ;
188 Logic Programming With Prolog
X = bill ,
Y = margaret ;
X = bill ,
Y = jane ;
X = george ,
Y = carol ;
X = george ,
Y = margaret ;
X = george ,
Y = jane ;
X = alfred ,
Y = carol ;
X = alfred ,
Y = margaret ;
X = alfred ,
Y = jane ;
no
Specimen Solutions to Practical Exercises 189
Practical Exercise 3
Question 1
Suitable definitions are given below:
child_of(A,B):-parent(B,A).
grandfather_of(A,B):-father(A,C),parent(C,B).
grandmother_of(A,B):-mother(A,C),parent(C,B).
great_grandfather_of(A,B):father(A,C),grandfather_of(C,B).
great_grandfather_of(A,B):father(A,C),grandmother_of(C,B).
?- child_of(X,ann).
X = henry ;
X = mary ;
no
?- grandfather_of(A,caroline).
A = francis ;
no
?- grandmother_of(B,caroline).
B = janice ;
no
?- great_grandfather_of(C,caroline).
C = john ;
no
Question 2
The system begins by matching the goal with the first clause defining the
ancestor/2 predicate, i.e. [A1].
190 Logic Programming With Prolog
?-ancestor(louise,Desc).
[A1] ancestor(louise,Y):-parent(louise,Y).
X is bound to louise. Variables Desc and Y are bound to each other.
The goal parent(louise,Y) is now matched with clause [P3], which is first
rewritten to replace X and Y by X1 and Y1, i.e.
parent(X1,Y1):-mother(X1,Y1).
?-ancestor(louise,Desc).
[A1] ancestor(louise,Y):-parent(louise,Y).
[P3] parent(louise,Y1):-mother(louise,Y1).
X is bound to louise. Variables Desc, Y and Y1 are bound to each other. X1 is
bound to louise.
The system now tries to satisfy the goal mother(louise,Y1). It matches it with
clause [M9].
?-ancestor(louise,Desc).
[A1] ancestor(louise,Y):-parent(louise,Y).
[P3] parent(louise,Y1):-mother(louise,Y1).
[M9] mother(louise,caroline).
X is bound to louise. Variables Desc, Y and Y1 are bound to each other and to
caroline. X1 is bound to louise.
This gives a solution to the user's goal, with Desc bound to caroline.
?- ancestor(louise,Desc).
Desc = caroline
If the user now forces the system to backtrack, the system will try to resatisfy the
goal mother(louise,Y1) and fail. This will cause the rule [P3] to be rejected.
Attempts to resatisfy parent(louise,Y) in the body of [A1] will also fail, so clause
[A1] will be rejected. This brings the system back to the original goal
ancestor(louise,Desc).
The system tries to resatisfy it by matching it with the second clause for
ancestor/2, i.e. [A2].
Specimen Solutions to Practical Exercises 191
?-ancestor(louise,Desc).
[A2] ancestor(louise,Y):-parent(louise,Z),ancestor(Z,Y).
X is bound to louise. Variables Desc and Y are bound to each other. Variable Z is
unbound.
The system now tries to satisfy the goal parent(louise,Z). It matches it with [P3],
which is first rewritten as (say)
parent(X1,Y1):-mother(X1,Y1).
?-ancestor(louise,Desc).
[A2] ancestor(louise,Y):-parent(louise,Z),ancestor(Z,Y).
[P3] parent(louise,Y1):-mother(louise,Y1).
X is bound to louise. Variables Desc and Y are bound to each other. Variables Y1
and Z are bound to each other. Variable X1 is bound to louise.
It now tries to satisfy the goal mother(louise,Y1). It matches it with clause [M9].
?-ancestor(louise,Desc).
[A2] ancestor(louise,Y):-parent(louise,caroline),ancestor(caroline,Y).
[P3] parent(louise,caroline):-mother(louise,caroline).
[M9] mother(louise,caroline).
X is bound to louise. Variables Desc and Y are bound to each other. Variables Z
any Y1 are bound to each other and to caroline. Variable X1 is bound to louise.
The next step is to satisfy the goal ancestor(caroline,Y). It is matched with [A1],
which is first rewritten as (say)
ancestor(X2,Y2):-parent(X2,Y2).
192 Logic Programming With Prolog
?-ancestor(louise,Desc).
[A2] ancestor(louise,Y):-parent(louise,caroline),ancestor(caroline,Y).
[P3] parent(louise,caroline):-mother(louise,caroline).
[M9] mother(louise,caroline).
[A1] ancestor(X2,Y2):-parent(caroline,Y2).
X is bound to louise. Variables Desc and Y are bound to each other. Variables Z
any Y1 are bound to each other and to caroline. Variable X1 is bound to louise.
Variable X2 is bound to caroline. Variables Y and Y2 are bound to each other.
Next the system tries to satisfy parent(caroline,Y2). It matches it with [P3], which
is first rewritten as (say)
parent(X3,Y3):-mother(X3,Y3).
?-ancestor(louise,Desc).
[A2] ancestor(louise,Y):-parent(louise,caroline),ancestor(caroline,Y).
[P3] parent(louise,caroline):-mother(louise,caroline).
[M9] mother(louise,caroline).
[A1] ancestor(X2,Y2):-parent(caroline,Y2).
[P3] parent(caroline,Y3):-mother(caroline,Y3).
X is bound to louise. Variables Desc and Y are bound to each other. Variables Z
any Y1 are bound to each other and to caroline. Variable X1 is bound to louise.
Variable X2 is bound to caroline. Variables Y, Y2 and Y3 are bound to each other.
X3 is bound to caroline.
Specimen Solutions to Practical Exercises 193
Next the system tries to satisfy mother(caroline,Y3). It matches it with [M10].
?-ancestor(louise,Desc).
[A2] ancestor(louise,Y):-parent(louise,caroline),ancestor(caroline,Y).
[P3] parent(louise,caroline):-mother(louise,caroline).
[M9] mother(louise,caroline).
[A1] ancestor(X2,Y2):-parent(caroline,Y2).
[P3] parent(caroline,Y3):-mother(caroline,Y3).
[M10] mother(caroline,david).
X is bound to louise. Variables Desc and Y are bound to each other. Variables Z
any Y1 are bound to each other and to caroline. Variable X1 is bound to louise.
Variable X2 is bound to caroline. X3 is bound to caroline. Variables Y, Y2 and Y3
are bound to each other and to david.
Now all the goals in the body of [A2] have succeeded, so the original goal
ancestor(louise,Desc) succeeds with Desc bound to david.
Another solution with Desc bound to janet is available by backtracking.
?- ancestor(louise,Desc).
Desc = caroline ;
Desc = david ;
Desc = janet ;
no
194 Logic Programming With Prolog
Practical Exercise 4
Question 1
The following program is a possible solution. There is no need to change the
names of the predicates to isa_dog etc., but it makes the program easier to read if
you do.
?-op(150,xf,isa_dog).
?-op(150,xf,isa_cat).
?-op(150,xf,is_large).
?-op(150,xf,is_small).
?-op(150,xf,isa_large_dog).
?-op(150,xf,isa_small_animal).
?-op(150,xfy,chases).
fido isa_dog. fido is_large.
mary isa_cat. mary is_large.
rover isa_dog. rover is_small.
jane isa_cat. jane is_small.
tom isa_dog. tom is_small.
harry isa_cat.
fred isa_dog. fred is_large.
henry isa_cat. henry is_large.
bill isa_cat.
steve isa_cat. steve is_large.
jim is_large.
mike is_large.
X isa_large_dog:- X isa_dog,X is_large.
A isa_small_animal:- A isa_dog,A is_small.
B isa_small_animal:- B isa_cat,B is_small.
X chases Y:X isa_large_dog,Y isa_small_animal,
write(X),write(' chases '),write(Y),nl.
Specimen Solutions to Practical Exercises 195
There are six possible ways of satisfying the goal X chases Y.
?- X chases Y.
fido chases rover
X = fido ,
Y = rover ;
fido chases tom
X = fido ,
Y = tom ;
fido chases jane
X = fido ,
Y = jane ;
fred chases rover
X = fred ,
Y = rover ;
fred chases tom
X = fred ,
Y = tom ;
fred chases jane
X = fred ,
Y = jane ;
no
Question 2
A possible definition is given below.
pred(A,B):-X is (A+B)/2,write('Average is: '),
write(X),nl,Y is sqrt(A*B),
write('Square root of product is: '),write(Y),nl,
Z is max(X,Y),write('Larger is: '),write(Z),nl.
?- pred(6,7).
Average is: 6.5
Square root of product is: 6.48074069840786
Larger is: 6.5
yes
196 Logic Programming With Prolog
?- pred(1,9).
Average is: 5
Square root of product is: 3
Larger is: 5
yes
Specimen Solutions to Practical Exercises 197
Practical Exercise 5
Question 1
A possible definition is as follows:
makelower:-get0(X),process(X).
process(13):-nl.
process(X):-X=\=13,convert(X,Y),put(Y),makelower.
convert(X,Y):-X>=65,X=<=90,Y is X+32.
convert(X,X):-X<65.
convert(X,X):-X>90.
Question 2
copyterms(Infile,Outfile):seeing(S),telling(T),
see(Infile),tell(Outfile),
copy,
seen,see(S),told,tell(T).
copy:-read(X),process(X).
process(end_of_file).
process(X):X\=end_of_file,
write(X),write('.'),nl,copy.
Note the use of write('.') to output a full stop after each term, so that the file may
subsequently be read in again as specified.
198 Logic Programming With Prolog
Question 3
A possible definition of readfile is given below.
readfile(F):seeing(S),see(F),
readchar,readchar,readchar,
readchar,readchar,readchar,
readchar,readchar,readchar,
readchar,readchar,readchar,
readchar,readchar,readchar,
seen,see(S).
readchar:-get0(X),write(X),nl.
If 'end of file' and 'end of record' are represented as suggested in Chapter 5, the
output from a call to readfile would be:
?- readfile('testa.txt').
97
98
99
100
101
13
10
102
103
104
105
106
13
10
-1
yes
If this is not the output produced on your system, it means that your version of
Prolog uses a different way of representing end of file, end of record or both.
Question 4
It will be assumed that fill in1.txt contains
Specimen Solutions to Practical Exercises 199
first.
second.
third.
fourth.
'fifth and last'.
end.
and file in2.txt contains
alpha.
beta.
gamma.
omega.
end.
A possible definition of combine is as follows.
combine(In1,In2,Out):seeing(S),telling(T),
tell(Out),see(In1),copyfile,seen,
see(In2),copyfile,seen,see(S),
write(end),nl,told,telling(T).
copyfile:-read(N),process(N).
process(end).
process(N):-write(N),nl,copyfile.
Testing this by
?- combine('in1.txt','in2.txt','out.txt').
produces the output file out.txt given below.
first
second
third
fourth
fifth and last
alpha
beta
gamma
omega
end
200 Logic Programming With Prolog
Question 5
It will be assumed that the file test1.txt contains
first.
second.
third.
fourth.
end.
and file test2.txt contains
first.
xxxx.
third.
fourthxxx.
end.
A possible definition of compare is:
compare(File1,File2):seeing(S),compfile(File1,File2),
see(File1),seen, see(File2),seen,see(S).
compfile(File1,File2):see(File1),read(X),see(File2),read(Y),
comp(X,Y),process(X,Y,File1,File2).
comp(A,A):-write(A),write(' is the same as '),
write(A),nl.
comp(A,B):-write(A),write(' is different from '),
write(B),nl.
process(end,end,_,_).
process(_,_,File1,File2):-compfile(File1,File2).
?- compare('test1.txt','test2.txt').
first is the same as first
second is different from xxxx
third is the same as third
fourth is different from fourthxxx
end is the same as end
yes
Specimen Solutions to Practical Exercises 201
Practical Exercise 6
Question 1
A possible definition using recursion is given below:
outsquare(N1,N2):-N1>N2.
outsquare(N1,N2):write(N1),write(' squared is '),Square is N1*N1,
write(Square),nl,M is N1+1,outsquare(M,N2).
?- outsquare(6,12).
6 squared is 36
7 squared is 49
8 squared is 64
9 squared is 81
10 squared is 100
11 squared is 121
12 squared is 144
yes
Question 2
The following definition uses the repeat predicate:
go:-repeat,get0(X),getrest(X).
getrest(13):-nl.
/* newline */
getrest(63):-nl,repeat,get0(X),X=:=13.
getrest(X):-put(X),fail.
?- go.
: abcdef
abcdef
yes
?- go.
: abcde?wxyz
abcde
yes
202 Logic Programming With Prolog
Question 3
The following program uses backtracking with failure:
find:person(_,_,Age,_,Prof),Age>40,
write('Profession is '),write(Prof),nl,fail.
find.
person(john,smith,45,london,doctor).
person(martin,williams,33,birmingham,teacher).
person(henry,smith,26,manchester,plumber).
person(jane,wilson,62,london,teacher).
person(mary,smith,29,glasgow,surveyor).
?- find.
Profession is doctor
Profession is teacher
yes
Specimen Solutions to Practical Exercises 203
Practical Exercise 7
Question 1
The corrected version is as follows.
factorial(1,1):-!.
factorial(N,Nfact):-N1 is N-1,
factorial(N1,Nfact1),Nfact is N*Nfact1.
?- factorial(6,N).
N = 720
?- factorial(7,M).
M = 5040
Question 2
A possible completed program is given below. It uses the same method of testing
whether a number is even that was used in Chapter 4, i.e. use // to divide the
number by 2, discarding any remainder, then multiply by 2 and check if the result
is the original number.
go:-repeat,read_and_check(N,Type),
write(N),write(' is '),write(Type),nl,N=:=100.
read_and_check(N,Type):write('Enter next number'),read(N),
check(N,Type).
check(N,even):-N1 is N//2,N2 is 2*N1,N2=:=N,!.
check(N,odd).
204 Logic Programming With Prolog
Practical Exercise 8
Question 1
A possible answer is given below. Predicate process adds a fact to the database
only if is not already there.
add_data:assert(animal(dummy)),
repeat,write('Enter next name'),
read(X),process(X),X=end,
retract(animal(dummy)).
process(end):-!.
process(X):-animal(X),write('Duplicate entry'),nl,!.
process(X):-assert(animal(X)),!.
Question 2
The display_animals predicate defined below uses backtracking with failure to list
the names of all the animals in the database.
display_animals:-animal(X),write(X),nl,fail.
display_animals.
Question 3
The remove predicate removes clauses from the database if they are present. If
they are not, it does nothing but still succeeds.
remove2:-remove(dog),remove(cat).
remove(X):-retract(animal(X)).
remove(X).
Specimen Solutions to Practical Exercises 205
Practical Exercise 9
Specimen solutions are given below. All the definitions require just one or two
clauses.
Question 1
pred1([A|L],L).
Question 2
inc([],[]).
inc([A|L],[A1|L1]):-A1 is A+1,inc(L,L1).
Question 3
palindrome(A):-reverse(A,A).
Question 4
putfirst(A,L,[A|L]).
Question 5
putlast(A,L,L1):-append(L,[A],L1).
Question 6
pred1(L1,L):-findall([A],member(A,L1),L).
pred2(L1,L):-findall(pred(A,A),member(A,L1),L).
pred3(L1,L):-findall([element,A],member(A,L1),L).
206 Logic Programming With Prolog
Practical Exercise 10
Specimen solutions are given below.
Question 1
spalindrome(S):-name(S,L),reverse(L,L).
Question 2
remove_final(S,S1):name(S,L),reverse(L,L1),
removespaces(L1,L2),reverse(L2,L3),
name(S1,L3).
removespaces([],[]):-!.
removespaces([32|L],L1):-removespaces(L,L1),!.
removespaces(L,L).
Question 3
replace(S1,S2):-name(S1,L1),rep(L1,L2),name(S2,L2).
rep([_|L],[63|L]).
Specimen Solutions to Practical Exercises 207
Practical Exercise 11
Question 1
One solution is to place the operator definitions
?-op(200,fx,head).
?-op(200,fx,tail).
near the beginning of the program file and the clauses
H iss head [H|T]:-!.
T iss tail [H|T]:-!.
anywhere amongst the clauses defining the iss predicate (as long as it is before the
final clause).
Question 2
Using the univ operator this predicate can be defined with just one clause.
addArg(Term1,NewArg,Term2):Term1=..L,append(L,[NewArg],L2),
Term2=..L2.
The original term is converted to a list, the extra term is added to the end of the list
using the append/3 predicate and the resulting list is converted back to a term.
?- addArg(person(john,smith,25),london,T).
T = person(john,smith,25,london)
?- addArg(city(paris,france),[a,b,c],T).
T = city(paris,france,[a,b,c])
The definition also works if the first argument is an atom, not a compound term. A
term with one argument is created.
?- addArg(height,200,T).
T = height(200)
Appendix 4
Glossary
(Words in bold are cross-references to other entries in the glossary.)
Anonymous Variable See Variable
Argument See Term
Arithmetic Expression A valid combination of numbers, variables,
arithmetic operators and arithmetic functions, for example
4.37+6*X-Y+sqrt(67.4).
Arithmetic Function A predicate such as sin, sqrt or abs used in an
arithmetic expression that (unlike predicates used elsewhere in
Prolog) returns a numerical value.
Arithmetic Operator An operator such as + - * / used in an arithmetic
expression that (unlike operators used elsewhere in Prolog) returns a
numerical value.
Arity See Term
ASCII Value of a Character An integer from 0 to 255 associated with
each of the (up to) 256 possible characters that may be used by the
Prolog system. See Chapter 5 (Section 5.3) for a table giving the
ASCII values of the most common characters.
210 Logic Programming With Prolog
Atom A non-numeric constant, e.g. dog or 'a long atom'
Backtracking The process of going back to a previous goal to find
alternative ways of satisfying it (see Chapter 3). Backtracking can be
prevented by using the cut predicate (see Chapter 7).
Backtracking with Failure A technique that can be used to search through
all the clauses in the Prolog database or to find all possible ways of
satisfying a goal (see Chapter 6).
Binary Predicate A predicate that has two arguments.
Binding a Variable The process of giving a value to a variable.
BIP Abbreviation for Built-in Predicate
Body of a Clause See Clause
Body of a Rule See Clause
Bound Variable One that has been given a value.
Built-in Predicate (BIP) See Predicate
Call Term A term that is either an atom or a compound term. Goals
entered by the user, heads of clauses and the components of bodies of
rules are all of this form.
Character One of a set of symbols that can be represented in a computer.
These can be letters, digits, spaces, punctuation marks etc. See also
ASCII Value of a Character and White Space Character
Glossary 211
Clause A Prolog program comprises a sequence of clauses. There are two
types of clause: facts and rules.
(1) Facts are of the form
head.
head is called the head of the clause. It may be an atom or a compound
term, whose functor is any atom (except :-). Some examples of facts are:
christmas.
likes(john,mary).
likes(X,prolog).
dog(fido).
(2) Rules are of the form:
head:-t1,t2, … , tk. (k>=1)
head is called the head of the clause (or the head of the rule).
It must be an atom or a compound term, whose functor is any atom (except
:-).
:- is called the neck of the clause (or the 'neck operator').
t1,t2, … , tk is called the body of the clause (or the body of the rule). It
consists of one or more goals, separated by commas.
The neck operator :- is read as 'if'. Commas are read as 'and'. Thus a rule
can be read as 'head is true if t1, t2, …, tk are all true'.
Some examples of rules are:
large_animal(X):-animal(X),large(X).
grandparent(X,Y):-father(X,Z),parent(Z,Y).
go:-write('hello world'),nl.
Closed World Assumption Any conclusion that cannot be proved to
follow from the facts and rules in the database is false. There is no
other information.
212 Logic Programming With Prolog
Compound Term See Term
Concatenating Lists Combining two lists to give a new list, the elements
of which are those of the first list followed by those of the second list.
For example, concatenating the lists [a,b,c,d] and [1,2,3] gives the list
[a,b,c,d,1,2,3].
Cons Character The vertical bar character | used to construct a list from its
head and tail (see Chapter 9).
Current Input Stream See Files
Current Output Stream See Files
Cut A special built-in predicate used to prevent backtracking (see Chapter
7).
Cut with failure A technique used to specify exceptions to general rules
(see Chapter 7).
Database The Prolog database comprises a set of clauses (rules and facts)
which constitute definitions for one or more predicates. Clauses are
generally loaded into the database from a text file by entering a
consult directive at the system prompt (see the built-in predicate
consult described in Chapter 1). Clauses can also be added to the
database as a side effect when a goal is evaluated (see the built-in
predicates asserta and assertz described in Chapter 8).
Clauses placed in the database remain there until one of the following
happens:
(a) One or more clauses are deleted as a side effect when a goal is evaluated
(see the retract and retractall built-in predicates described in Chapter 8).
(b) Further clauses are loaded into the database from a text file (if these
include one or more clauses for a predicate already stored in the database,
all the previously stored clauses for that predicate are first automatically
deleted).
Glossary 213
(c) The user exits from the Prolog interpreter (all clauses are automatically
deleted).
Declarative Interpretation of a Rule Rules have both a declarative and a
procedural interpretation. The declarative interpretation is that its
head is satisfied if all the goals in its body are satisfied. With this
reading, the order of the clauses defining a predicate and the order of
the goals in the body of each rule are irrelevant. See also Procedural
Interpretation of a Rule.
Declarative Program A declarative program is one in which the order of
the clauses defining each predicate and the order of the goals in the
body of each rule do not affect the answers to a user query (including
multiple answers produced by backtracking). This aim may be either
fully or partly achieved. It is considered to be good Prolog
programming practice to make programs declarative as far as possible
(see Section 3.6).
Directive A goal included in a Prolog program, prefixed by the system
prompt characters (see Section 4.1).
Disjunction Operator The disjunction operator ;/2 (written as a semicolon
character) is used to represent 'or'. It is an infix operator that takes two
arguments, both of which are goals. Goal1;Goal2 succeeds if either
Goal1 or Goal2 succeeds.
Empty List See List
Equality Operator An operator used for testing the equality of two
arithmetic expressions or two terms.
Evaluate a Goal Determine whether or not a goal is satisfied.
Existentially Quantified Variable See Variable
Fact A fact is a type of clause.
214 Logic Programming With Prolog
Files The same facilities available for input and output from and to the
user's terminal are available for input and output from and to files
(e.g. on a hard disk or a CD-ROM).
Prolog takes all input from the current input stream and writes all output to
the current output stream. The user may open and close input streams and
output streams associated with any number of named files but there can
only be one current input stream and one current output stream at any time.
Function A relationship between a number of values, such as 6+4 or
sqrt(N), which evaluates to a number (or potentially some other kind
of term), rather than to true or false as for predicates. Prolog does not
make use of functions except in arithmetic expressions (see Chapter
4).
Functor See Term
Goal A component of a query entered by the user at the system prompt,
such as go, animal(X) or factorial(6,M), which either succeeds or fails.
The head of a clause can be viewed as a goal with, in the case of a
rule, the components of its body as subgoals. The components in the
body of a rule are also known as goals. Wherever they appear, goals
always take the form of call terms.
Head of a Clause See Clause
Head of a List See List
Head of a Rule See Clause
Infix Operator A predicate with two arguments written using a special
notation where the functor is placed between its two arguments with
no parentheses, e.g. john likes mary. See also Operator.
Input Stream See Files
Glossary 215
Lexical Scope of a Variable If a variable, say X, appears in a clause, it is
entirely different from any variable named X that may appear in
another clause. This is expressed by saying that the lexical scope of a
variable is the clause in which it appears.
List A type of compound term, which is written not in the usual 'functor
and argument' notation, but as an unlimited number of arguments
(known as list elements) enclosed in square brackets and separated by
commas, e.g. [dog,cat,fish,man].
An element of a list may be a term of any kind, including another list, e.g.
[x,y,mypred(a,b,c),[p,q,r],z]
[[john,28],[mary,56,teacher],robert,parent(victoria,albert),
[a,B,[C,D,e],f],29]
A list with no elements is known as the empty list. It is written as [].
The first element of a non-empty list is called its head. The list remaining
after the head is removed is called the tail of the original list.
For example the head of the list [x,y,mypred(a,b,c),[p,q,r],z] is the term x
(an atom) and the tail is the list [y,mypred(a,b,c),[p,q,r],z].
The head of the list
[[john,28],[mary,56,teacher],robert,parent(victoria,albert),
[a,B,[C,D,e],f],29]
is the term [john,28] (which is a list). The tail is the list
[[mary,56,teacher],robert,parent(victoria,albert),[a,B,[C,D,e],f],29].
List Element See List
List Processing Performing operations on the elements of one or more
lists. See Chapter 9.
Logic Programming A style of programming derived from research in the
field of computational logic. It is most commonly embodied in the
programming language Prolog (Programming in Logic). The clauses
216 Logic Programming With Prolog
in a Prolog program have a close similarity to propositions in
mathematical logic.
Looping Evaluating a set of goals repeatedly either a fixed number of times
or until a condition is met (see Chapter 6).
Neck of a Clause See Clause
Neck Operator See Clause
Operator A predicate with two arguments can be converted to an infix
operator in the interests of readability of programs. The functor is
written between the two arguments, e.g. john likes mary instead of
likes(john,mary). A predicate with one argument can be converted to
either a prefix operator or a postfix operator. The functor is written
either before (prefix) or after (postfix) the argument, e.g. isa_dog fred
or fred isa_dog instead of isa_dog(fred).
Arithmetic Operator An operator such as + - * / used in an arithmetic
expression that (unlike operators used elsewhere in Prolog) returns a
numerical value.
Equality Operator An operator used for testing the equality of two
arithmetic expressions or two terms.
Relational Operator An operator used for comparing numerical values,
such as < denoting 'less than'.
Operator Precedence A number (also called the precedence value)
associated with an operator that determines the order in which
operators will be applied when more than one is used in a term (see
Chapter 4).
Output Stream See Files
Postfix Operator A predicate with one argument written using a special
notation where the functor is placed after its argument with no
parentheses, e.g. fred isa_dog. See also Operator.
Glossary 217
Precedence Value See Operator Precedence
Predicate All the clauses in the Prolog database for which the head has the
same combination of functor and arity comprise the definition of a
predicate. Predicates are sometimes described using functor and arity
notation, e.g. write/1.
Binary Predicate: a predicate that has two arguments
Unary Predicate: a predicate that has one argument
Built-in Predicate: a standard predicate defined by the Prolog system.
Prefix Operator A predicate with one argument written using a special
notation where the functor is placed before its argument with no
parentheses, e.g. isa_dog fred. See also Operator.
Procedural Interpretation of a Rule Rules have both a declarative and a
procedural interpretation. The procedural interpretation is that in
order to satisfy its head each of the goals in its body should be
satisfied in turn, working from left to right. With this reading, the
order of the goals in the body of each rule and the order of the clauses
defining a predicate are of great importance. See also Declarative
Interpretation of a Rule.
Program A Prolog program comprises the clauses (rules and facts)
currently held in the Prolog database. Unlike most other
programming languages there is no fixed way in which a program
must be used. Entering a goal or a sequence of goals in response to the
system prompt causes the Prolog interpreter to search through the
clauses relevant to satisfying that goal or goals, as described in
Chapter 4.
Prolog A programming language embodying the ideas of logic
programming.
Prompt An abbreviated form of System Prompt
218 Logic Programming With Prolog
Query A sequence of one or more goals entered by the user at the prompt.
In this book the more explicit term 'sequence of goals' is generally
used.
Re-evaluate a Goal Determine whether or not a goal can be resatisfied.
Recursive Definition of a Predicate One that uses the predicate itself,
either directly or indirectly.
Relational Operator An operator used for comparing values, such as <
denoting 'less than'.
Resatisfy a Goal Find another way of satisfying a goal, whilst
backtracking.
Rule A rule is a type of clause.
Satisfy a Goal Prove that the goal follows from the facts and rules in the
database. This usually involves binding one or more variables (see
Chapter 3).
Sequence of Goals A combination of goals joined together by commas,
signifying 'and'. In order for the sequence of goals to succeed, all the
individual goals must succeed.
Side Effect An action taken by the Prolog system, such as writing a line of
text or opening a file, whilst attempting to satisfy a goal.
String A collection of characters such as 'hello world'. An atom can be
regarded as a string (see Chapter 10).
String Processing Performing operations on the contents of one or more
strings (see Chapter 10).
Glossary 219
Subgoal See goal.
Sublist A list element that is itself a list.
System Prompt A combination of characters (?- in this book) output by
the Prolog system to indicate that it is ready for the user to enter a
sequence of one or more goals.
Tail of a List See List
Term The name given to the data objects in Prolog. A term can be an
atom, a variable, a number, a compound term or a list. Some dialects
of Prolog allow other possibilities, e.g. strings.
A compound term is a structured data type that consists of a functor
followed by a sequence of one or more arguments, which are enclosed in
brackets and separated by commas. The general form is: functor(t1,t2, … ,tn)
n≥1.
The functor must be an atom. Each argument must be a term (possibly a
compound term). The number of arguments a compound term has is called
its arity. A compound term can be thought of as representing a record
structure. The functor represents the name of the record, while the
arguments represent the record fields.
A call term is an atom or a compound term. The body of a rule consists of a
sequence of call terms, separated by commas.
Unary Predicate A predicate that has one argument. See also Predicate
Unbound Variable One that does not have a value.
Unification A process of matching, generally involving binding variables,
to make two call terms identical (see Chapter 3).
Universally Quantified Variable See Variable
220 Logic Programming With Prolog
Unresatisfiable Goal One that always fails when backtracking.
User's Terminal A generic term that normally refers to the user's keyboard
(for input) and screen (for output). See also Files
Variable In a query a variable is a name used to stand for a term that is to
be determined, e.g. variable X may stand for atom dog, the number 12.3, a
compound term or a list. The meaning of a variable when used in a rule or
fact is described in Chapter 2. See also Lexical Scope of a Variable
Unbound Variable: one that does not have a value.
Bound Variable: one that has been given a value. A bound variable may
become unbound again and possibly then bound to a different value by the
process of backtracking, described in Chapter 3.
Universally Quantified Variable: one that appears in the head of a clause,
indicating that the corresponding fact or rule applies for all possible values
of the variable.
Existentially Quantified Variable: One that appears in the body of a rule,
but not in its head, signifying that 'there exists at least one value of the
variable'.
Anonymous Variable: A variable used in a fact, rule or goal entered by the
user when the value is unimportant (see Chapter 2).
White Space Character A non-printing character, such as space or tab.
Formally, a character with an ASCII value less than or equal to 32.
Index
Page numbers in bold refer to definitions in the Glossary (Appendix 4). Page
numbers in bold italic refer to definitions in the lists of Built-in Predicates and
Built-in Operators (Appendices 1 and 2).
! See 'Cut' Predicate
, [comma] Operator 179
; [semicolon] Operator 66-67, 179
\= Operator 65, 180
\== Operator 64, 180
< Operator 62, 182
= [equals] Operator 64, 180
=.. ['univ'] Operator 160, 181
=:= Operator 62, 63, 181
=\= Operator 62, 63, 181
=< Operator 62, 182
== Operator 64, 180
> Operator 62, 182
>= Operator 62, 182
Anonymous Variable 10, 25-26, 220
append Predicate 127, 131-132, 167
Append to a File 78
Applications of Prolog viii
arg Predicate 162-163, 168
Argument 4, 10, 18, 219
Arithmetic Expression 59, 60, 62,
63, 64, 209
Arithmetic Function 60, 209
Arithmetic in Prolog 59-62, 149-155
Arithmetic Operator 60, 149, 209
Arity 10, 15, 16, 30, 219
ASCII Value of a Character 72-74,
138, 209
asserta Predicate 110-111, 168
assertz Predicate 110, 168
Atom 4, 9, 11, 14, 15, 16, 17, 210
atom Predicate 156, 168
atomic Predicate 169
Backtracking 7, 23, 31, 39-49, 9196, 99, 210
Backtracking with Failure 94-96,
210
Binary Predicate 57, 210
Binding a Variable 23, 31, 210
BIP See Built-in Predicate
Body of a Clause 14, 211
Body of a Rule 14, 211
Bound Variable 23, 210
Built-in Predicate (BIP) 2, 17, 59,
217
call Predicate 161, 169
Call Term 11, 13, 14, 30, 31, 32-35,
210
Character 72-77, 210
Clause 3, 4, 13-15, 211
Closed World Assumption 31, 211
Compound Term 10-11, 219
Concatenating Lists 127-129, 212
Cons Character 120-124, 212
consult Predicate 4, 19-21, 109, 169
Current Input Stream 71, 73, 77, 214
Current Output Stream 17, 69-70,
73, 77, 214
'Cut' Predicate 99-107, 167, 212
222 Logic Programming With Prolog
Cut with failure 106-107, 212
Database 4, 109-117, 212
Declarative Interpretation of a Rule
vi, 16-17, 213 See also Procedural
Interpretation of a Rule
Declarative Program 52-54, 213
Directive 59, 213
Disjunction Operator 66-67, 213
dynamic Predicate 110, 169
Empty List 119, 215
Equality Operators 63-66, 213
Evaluate a Goal 5, 35-51, 213
Existentially Quantified Variable 2425, 220
Expert System viii
Fact vi, vii, 4, 13, 14, 114, 213
fail Predicate 94, 107, 169
Files 77-81, 214
findall Predicate 133-135, 170
Function 18, 214
Functor 10, 15, 16, 17, 18, 30, 219
functor Predicate 162, 170
get Predicate 73, 78, 141, 170
get0 Predicate 73, 78, 141, 170
Goal 1, 2, 3, 21-23, 30, 214
halt Predicate 2,3, 171
Head of a Clause 13, 211
Head of a List 120, 215
Head of a Rule 13, 14, 211
List Processing ix, 119-135, 215
listing Predicate 7-8, 171
Loading Clauses 19-21, 109
Logic Programming v, 215
Logical Operators 66-67
Looping 85-96, 216
member Predicate 125, 131, 172
name Predicate 137-138, 172
Neck of a Clause 14, 211
Neck Operator 14, 211
nl Predicate 2, 172
not Operator 66, 182
op Predicate 58-59, 172
Operator 57-59, 62, 63-67, 149, 216
Operator Precedence 58, 62, 216
Output Stream 17, 69-70, 77, 214
Postfix Operator 58, 216
Precedence Value See Operator
Precedence
Predicate 2, 4, 15-18, 30, 59, 217
Prefix Operator 57, 217
Procedural Interpretation of a Rule
vi, 16-17, 217 See also
Declarative Interpretation of a
Rule
Program v, 217
Prolog v-x, 217
Prompt See System Prompt
put Predicate 73, 173
Query vi, vii, 3, 218
Infix Operator 57, 214
Input Stream 71, 73-74, 77, 78-79,
214
integer Predicate 171
is Operator 59, 182
length Predicate 126, 171
Lexical Scope of a Variable 23-24,
215
List 11, 119-124, 215
List Constructor Notation 120-124
List Element 11, 119, 215
Re-evaluate a Goal 51, 218
read Predicate 71, 78, 173
reconsult Predicate 19-21, 109, 173
Recursion ix, 18, 85-90, 129-132
Recursive Definition of a Predicate
18, 218
Relational Operator 62, 63-66, 218
repeat Predicate 91-93, 173
Resatisfy a Goal 39, 42-44, 218
retract Predicate 111, 174
retractall Predicate 111, 174
Index
reverse Predicate 126, 130, 131, 174
Rule vi, vii, 5, 13, 14, 16, 218
Satisfy a Goal 5, 29-31, 39-51, 96,
218
see Predicate 78, 80, 174
seeing Predicate 78, 80, 175
seen Predicate 78, 80, 175
Sequence of Goals 3, 218
Set Operations in Prolog 157-160
Side Effect 17, 19, 29, 37, 218
statistics Predicate 2, 3, 175
String 137, 218
String Processing 11, 137-146, 155157, 218
Subgoal 14, 135, 214
Sublist 119, 219
Syntax ix
System Prompt vi, 1, 2, 7, 219
Tail of a List 120, 215
tell Predicate 77, 80, 175
telling Predicate 77, 80, 176
223
Term 8-11, 219
Term Equality 64-66
Terminating Condition 86
Theorem Proving vii
told Predicate 77, 80, 176
true Predicate 176
Unary Predicate 57, 219
Unbound Variable 23, 31, 219
Unification 31-35, 52, 61, 64, 65,
163-165, 219
Universally Quantified Variable 24,
220
Unresatisfiable Goal 42, 69, 220
User-defined Predicates 17
User's Terminal 77, 220
Variable 5, 10, 21-26, 31, 220
Versions of Prolog ix-x
White Space Character 13, 71, 220
write Predicate 2, 69, 176
writeq Predicate 70, 177